Working Paper No. 266

Delegating Performance Evaluation

Igor Letina, Shuo Liu and Nick Netzer

First version: November 2016 This version: November 2018

University of Zurich

Department of Economics

Working Paper Series

ISSN 1664-7041 (print) ISSN 1664-705X (online)

Delegating Performance Evaluation

Igor Letina, Shuo Liu and Nick Netzer∗

This Version: November 2018First Version: November 2016

Abstract

We study optimal incentive contracts with multiple agents when performance eval-uation is delegated to a reviewer. The reviewer may be biased in favor of the agents,but the degree of bias is unknown to the principal. We show that a contest, whichis a contract in which the principal determines a set of prizes to be allocated to theagents, is optimal. By using a contest, the principal can commit to sustaining incen-tives despite the reviewer’s potential leniency bias. The optimal effort profile can beuniquely implemented by an all-pay auction with a cap. Our analysis has implica-tions for applications as diverse as the design of worker compensation, the awardingof research grants, and the allocation of foreign aid.

Keywords: performance evaluation, delegation, optimality of contestsJEL: D02, D82, M52

∗Letina: Department of Economics, University of Bern, Switzerland. Liu and Netzer: De-partment of Economics, University of Zurich, Switzerland. Email: igor.letina@vwi.unibe.ch,shuo.liu@econ.uzh.ch, and nick.netzer@econ.uzh.ch. We are grateful for useful comments and sug-gestions to Jean-Michel Benkert, Elchanan Ben-Porath, Lorenzo Casaburi, Yeon-Koo Che, DavidDorn, Christian Ewerhart, Alexander Frankel, William Fuchs, Sergiu Hart, Andreas Hefti, Jo-hannes Hörner, Navin Kartik, Christian Kellner, Botond Kőszegi, Wladislaw Mill, Harry Pei, RonSiegel, Florian Scheuer, Armin Schmutzler, Philipp Strack, Dezsö Szalay, and seminar participantsat Bar Ilan University, Bielefeld University, Carnegie Mellon University, Columbia University,Hebrew University of Jerusalem, LUISS University Rome, Northwestern University, Sun Yat-SenUniversity, Toulouse School of Economics, the Universities of Bern, Bonn, Cologne, Konstanz, andZurich, 18th Annual SAET Conference, 20th Colloquium on Personnel Economics, 28th Interna-tional Conference on Game Theory, 2018 Asian Meeting of the Econometric Society, 2018 ChinaMeeting of the Econometric Society, Organizational Economics Workshop Konstanz 2018, SIOEAnnual Conference 2017, SIRE Workshop on Behavioural Economics and Mechanism Design 2016,Swiss Economists Abroad Meeting 2016, Swiss Theory Day 2016, Verein für Socialpolitik Meeting2016, Verein für Socialpolitik Theoretischer Ausschuss 2017, Zurich Workshop in Economics 2016,and Zurich Workshop on Applied Behavioral Theory 2017. Shuo Liu would like to acknowledgethe hospitality of Columbia University, where some of this work was carried out, and financialsupport by the Swiss National Science Foundation (Doc.Mobility grant P1ZHP1_168260).

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1 Introduction

Principals often lack the information or expertise needed to make appropriate decisions.A common response to this problem is to delegate the decision to a better informed party.For example, funding agencies delegate the choice of research projects which will be fundedto an expert committee. Within a firm, the CEO usually delegates to a mid-level managerthe decision regarding the assignment of bonuses to subordinates. Humanitarian aid isdistributed by specialized agencies on behalf of the donor countries.

If the preferences of the principal and the expert who makes the decision are not aligned,then delegation can lead to distorted decisions. The principal can attempt to influence thedecision by limiting the set of outcomes from which the expert can select. The existingliterature on optimal delegation studies how this delegation set should be designed.1 Inthese papers, the principal wants to base her decision on some stochastic state of nature,the value of which is known only to the expert. Crucially, this state of nature is assumedto be exogenous. However, in the examples above, the state of nature (the quality ofresearch projects, the performance of employees, the cooperativeness of receiving countries)is determined in part in anticipation of the decision that the expert will make. As a matterof fact, the goal of the principal is exactly to incentivize the agents to exert effort. Forexample, the goal of the funding agencies is to stimulate creation of high quality research.Similarly, bonuses in firms are instruments that incentivize employees to work hard. Foreignaid is in part allocated to bring about reforms.

In this paper, we study the optimal delegation problem with an endogenous state ofnature. A principal wishes to incentivize agents to exert costly effort. The efforts are notobservable to the principal. However, an expert, which we will from now on refer to asthe reviewer, can (costlessly) observe the exerted efforts.2 The principal thus delegatesthe decision on how to reward the agents to the reviewer, but possibly restricts the setof allowable decisions. The reviewer’s preferences may not be perfectly aligned with theprincipal. While the reviewer takes into account the effect of his actions on the principal’spayoff, maybe because he owns shares of the company or he cares intrinsically, he may alsocare about the agents. For instance, as we will discuss below, there is ample evidence thatmanagers care about the payoffs of their subordinates. Exactly how much the reviewer

1See, for example, Holmström (1977, 1984), Melumad and Shibano (1991), Alonso and Matouschek(2008), Armstrong and Vickers (2010), Amador and Bagwell (2013), and Frankel (2014).

2We do not consider the problem of incentivizing the reviewer to exert costly effort in order to learnthe state of nature. This is an interesting but distinct incentive problem which is studied in Aghion andTirole (1997), Strausz (1997), Szalay (2005), Rahman (2012), and Pei (2015b). We also do not examinewhy the principal does not perform the task of the reviewer herself. We treat the existence of the revieweras given in the same way that we treat the existence of the agents are given. In the applications we havein mind, the principal often interacts simultaneously with many reviewers and agents. General Electric,which we will discuss again later on, has around 313,000 employees. It seems unlikely that GE’s CEO canin any reasonable manner monitor individual teams of workers. Furthermore, Strausz (1997) shows that itcan be optimal to delegate evaluation to a reviewer even when the principal could monitor an agent herself.

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cares about the agents is the reviewer’s private information. Importantly, a reviewer whocares sufficiently much about the agents will be reluctant to punish them even if they donot exert sufficient effort. Anticipating this, the agents will exert less effort. The principalthus has to design the delegation set in a way that restricts the scope of possible leniencyof the reviewer.

Of course, one could also imagine that the principal tries to correct the distortions bypaying transfers to the reviewer conditional on the action that he takes. Our paper belongsto the literature on optimal delegation where, like in models of cheap talk (Crawford andSobel, 1982) or bayesian persuasion (Kamenica and Gentzkow, 2011), conditional monetarytransfers are ruled out. While the delegation approach restricts the class of mechanisms overwhich the principal optimizes, we believe that this restriction is particularly plausible in oursetting of performance evaluation. A conditional transfer in this case would mean payinga reviewer more whenever he submits a more negative review. This is typically considereda conflict of interest and is either explicitly illegal or at least judged as repugnant.3

Our first main result is that a contest among the agents is an optimal delegation mech-anism. That is, the principal defines a set of prizes and the reviewer only decides how toallocate these prizes to the agents. The reviewer does not have the additional freedom tochoose the overall size or the split of the agents’ compensation. This strongly limits thedegree of leniency he can exercise. In particular, the reviewer is always forced to punishsome agents by assigning them a small prize, which is crucial for the preservation of in-centives. Without this commitment, the reviewer would be lenient and the agents wouldshirk. The downside of the contest mechanism is that a small prize has to be assigned (atrandom) even when all agents provide the sufficient level of effort. When the agents arerisk-averse, this will be inefficient.

This result is interesting for several reasons. First, contests are a commonly used andoften-studied incentive scheme, and much work has been devoted to their optimal design.4

There is, however, not much work on the more general question whether and under whichconditions contests are actually optimal in some class of mechanisms.5 Exceptions are theseminal paper of Lazear and Rosen (1981), as well as papers which stress that contests canfilter out common shocks when agents are risk-averse (Green and Stokey, 1983; Nalebuffand Stiglitz, 1983) or ambiguity-averse (Kellner, 2015). In our model, an important reason

3Another instrument that the delegation approach rules out in our setting is the possibility of directcommunication between the principal and the agents.

4For instance, Glazer and Hassin (1988) and Moldovanu and Sela (2001) study the design of prizes fora given contest success function. Jia, Skaperdas, and Vaidya (2013) survey papers that study the designof a contest success function for given prizes.

5Prendergast (1999, p. 36) writes: “Rather surprisingly, there is very little work devoted to understand-ing why this is the case, i.e., why the optimal means of providing incentives within large firms (at leastfor white-collar workers) seems to be tournaments rather than the other means suggested in the previoussections.” We find this still to be the case in the years since Prendergast published his paper.

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for contests to be optimal is that they act as a commitment device. A contest provides twotypes of commitment. It commits the principal to the announced prizes and thus preventsmanipulation of the sum of payments made to the agents. The literature has observedpreviously that this “commitment to pay” can be beneficial when the agents’ efforts arenot verifiable. For instance, Malcomson (1984, 1986) argues that piece-rate contracts arenot credible in that case, as the principal would always claim low performance ex post inorder to reduce payments, while a contest remains credible.6 However, this credibility canalso be achieved by simply committing to a total sum of payments without setting fixedprizes. In fact, as we will show, such a scheme would outperform a contest when the agentsare risk-averse, by removing uncertainty from equilibrium payments.7 Hence the secondtype of commitment, the above described “commitment to punish,” is crucial in explainingthe optimality of contests with fixed prizes.8 Second, a contest is a remarkably simplemechanism. Even though we allow for arbitrary stochastic delegation mechanisms withpossibly sophisticated transfer rules, the optimum can be achieved by a simple mechanismcharacterized by a prize profile and a suggestion how to distribute the prizes in response tothe agents’ efforts. The principal does not attempt to screen the reviewer’s type, in spiteof the fact that the first-best may be achievable if the type were known to the principal.This makes the strategic considerations of the agents simple. In particular, their behaviordoes not depend on beliefs regarding the reviewer’s type. This robustness property isimportant because principal and agents may well have different beliefs (for instance, inthe example with managers allocating bonuses, it seems reasonable to assume that theemployees working directly with a manager have more precise information about theirmanager’s type than a CEO does).

Our second main result characterizes the prize structure of an optimal contest. Givenn agents, an optimal contest will have n − 1 equal positive prizes and one zero prize.Thus, while the contest acts as a commitment to punish, the punishment is kept at theminimum required to incentivize effort. The delegation set forces the reviewer to allocatea zero prize to only one agent, so that the optimal contest exhibits a “loser-takes-nothing”

6Similarly, Carmichael (1983) considers a setting where the final output is verifiable but depends onthe efforts of both the principal and the agents. With a contract that pays agents based on total output,the principal has an incentive to reduce own effort in order to reduce the payments to agents.

7Levin (2002) shows that a constant sum of payments arises in equilibrium of a repeated game, wherethe principal’s wage promises have to be self-enforcing. A contest-type compensation structure arises inan optimal equilibrium only if the agents are risk-neutral.

8The problem of committing to punishment is related to Konrad (2001) and Netzer and Scheuer (2010).They study the problem of a planner who would like to implement redistribution after agents have chosentheir actions, the anticipation of which may destroy incentives to choose costly but socially desirableactions. In the context of optimal income taxation, Konrad (2001) shows that private information aboutlabor productivity provides a commitment against excessive redistribution. In the context of insuranceand labor markets, Netzer and Scheuer (2010) show that adverse selection provides a commitment bygenerating separating market equilibria. In both cases, agents who choose socially less desirable actionsare punished by having to forego information rents.

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rather than a “winner-takes-all” structure. In equilibrium, when all agents have providedsufficient effort, the reviewer randomly chooses the agent who receives the zero prize. Thusall agents are facing the same risk. If agents are risk-averse, they respond to this risk byreducing the amount of effort they are willing to exert. A corollary of this result is that thefirst-best is implementable if and only if the agents are risk-neutral. We also show that anoptimal contest implements an outcome close to the first-best if the agents’ risk-aversionis moderate or the number of agents is large.

Our third main result shows that a familiar all-pay auction with a cap implementsthe optimum, and it does so in unique equilibrium. As in a standard all-pay auctionwith n − 1 identical prizes, the agent with the lowest effort receives the zero prize. Tiesare broken randomly. However, efforts are capped at the desired equilibrium level. Thisremoves the possibility for the agents to exert slightly more than the equilibrium effortin order to guarantee themselves a positive prize with probability one. In addition to theall-pay auction, it can be shown that the optimum can also be achieved with an imperfectlydiscriminating contest, such as the well-known Tullock contest. Thus, the essential featureof our main result is the fixed profile of prizes, and not the exact form of the contest successfunction.

We then consider an extension where the reviewer only imperfectly observes the effortsof the agents. In a framework that allows for very general observational structures, we showthat the principal can often still implement the optimal allocation with a contest. Next, wediscuss how our analysis can be extended to settings with non-separable preferences andfavoritism. We also generalize the model to allow for heterogeneous effort cost functionsand show that our main results remain robust.

Our contribution is related to three strands of literature: on the optimality of contests,on biased reviewers, and on optimal delegation. A more detailed discussion of this literatureis postponed to Section 5. A closely related paper is Frankel (2014). Like in our paper,he considers a multidimensional delegation problem with uncertainty about the expert’spreferences. He assumes that the state of nature is exogenous and not affected by thechoice of the delegation mechanism. In contrast, our prime concern is how the delegationmechanism affects the state of nature, i.e., how it provides incentives for agents to exerteffort. Another difference is that Frankel (2014) derives max-min mechanisms, which areoptimal for the worst possible realization of the expert’s bias, while we are interestedin mechanisms that maximize the principal’s expected payoff given her beliefs about theexpert’s type. Frankel (2014) shows that, when the set of possible preferences of the expertis rich enough, a ranking mechanism is max-min optimal. When the multidimensional stateof nature is an effort profile, like in our setting, this ranking mechanism would correspondto a standard all-pay auction (see Siegel, 2009, for a general treatment of contests withall-pay structure). Such an all-pay auction without a cap is not optimal in our setting with

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endogenous efforts, but there is a range of different contests that are optimal. Furthermore,since the principal’s payoff turns out to be independent of the reviewer’s type in theseoptimal contests, they not only maximize expected payoffs but are also max-min optimal.Another related paper is Gennaioli (2013), who studies optimal contracting between twoparties when the contracts are executed by a potentially biased judge. The judge playsa role similar to the reviewer in our model, and a contract is similar to a delegation set(since the judge chooses an outcome from the menu of outcomes which are written in thecontract). Gennaioli (2013) shows that the judicial bias leads to contracts which are lesscontingent than the first-best. In contrast to our model, the distribution of the state ofnature in Gennaioli (2013) does again not depend on the choice of the contract. Gennaioliand Ponzetto (2017) do consider how contracting affects the provision of effort when judgesare biased, but their setting differs significantly from ours. In particular, they consider thecase of a single agent, while we study the case of multiple agents.9

Our model applies to many situations where a principal wants to incentivize agentsbut cannot directly supervise them. Here we will discuss two possible applications, whichare meant to illustrate the range and scale of our model. One application is the design ofperformance evaluation schemes in firms. The performance evaluation scheme is designedby the CEO, but the CEO does not observe the individual efforts of the employees towhich the scheme applies. Hence, the actual performance evaluation is delegated to theemployees’ supervisor. By virtue of working closely with the employees, the supervisorobserves their efforts but also cares about their payoffs. There is ample evidence (bothempirical and experimental) that supervisors tend to be too lenient when judging theperformance of their subordinates, and that the degree of leniency varies and dependson (among other things) social ties between the supervisor and the team.10 Our resultshave direct implications for the controversial debate over the use of the so-called “forcedrankings,” a review system which was most famously used by General Electric under JackWelch during their fast growth in the 1980s and 90s.11 Our contribution to this discussionis (i) to show that forced rankings are optimal for motivating effort under the assumptionsof our model, (ii) to show how optimal forced rankings should be constructed, and (iii)

to show that some elements of forced rankings which are usually criticized are actuallynecessary for incentivizing effort. In particular, forced rankings are criticized for forcing

9Another difference is that, in their setting, the principal and the agent need to present evidence whichthe judge can then verify or discard. This cannot be mapped into the standard delegation setting.

10For example, Bol (2011) and Breuer, Nieken, and Sliwka (2013) find evidence of leniency bias whichdepends on the strength of the employee-manager relationship. Bol (2011) cites studies documentingleniency bias going back to the 1920s, while citations to similar findings in the 1940s can be found inPrendergast (1999). Berger, Harbring, and Sliwka (2013) find experimental evidence of leniency bias,and Bernardin, Cooke, and Villanova (2000) document that the degree of leniency bias in an experimentdepends on personality traits of the reviewer.

11For example, see “‘Rank and Yank’ Retains Vocal Fans”, L. Kwoh, The Wall Street Journal, January31, 2012, and “For Whom the Bell Curve Tolls”, J. McGregor, The Washington Post, November 20, 2013.

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managers to assign low rankings even when all workers are performing well: “What happensif you’re working with a superstar team? You’ve just forced a distribution that doesn’texist. You create this stupid world where [great] people are punished.”12 Similarly, BradSmart who worked with Jack Welch on developing GE’s forced ranking system criticizedGE’s decision to assign 10% of the workers a low evaluation: “To force those distributionswhen the percentages don’t meet the reality is nuts.”13 Our results show that, far frombeing “stupid” or “nuts,” not rewarding some workers even when they perform well isnecessary, since if the managers were given an option to reward all workers, they maychoose it irrespective of actual performance, which would destroy any incentive effect ofthe evaluation system.

A very different situation for which our model offers insights is foreign aid. Donors havebeen trying for decades to use foreign aid to incentivize reforms in recipient countries, butthere is little empirical evidence that it has been effective (see e.g. Easterly, 2003; Rajanand Subramanian, 2008). In response, funding agencies and governments have tried toimprove mechanisms for the allocation of foreign aid in ways that link aid to improvementin governance and other policy reforms. One early approach has been the so-called “con-ditional aid,” where donors promise to withdraw future aid if the agreed policy reformshave not been achieved. However, the donors’ threats to withdraw aid were not credibleand, unsurprisingly, reforms were usually not carried out. As Easterly (2009) somewhatamusingly points out, the World Bank conditioned aid on the same agricultural policyreform in Kenya five separate times – and the conditions were violated each time. Svens-son (2003) proposes a solution to this problem. Instead of allocating the budget for eachcountry separately, similar countries could be pooled together and the total budget for allthese countries could be allocated to a single aid officer. This way, if one country doesnot reform, the aid officer has the option of reallocating the aid from that country to an-other. Our paper points to a potential problem with this approach and offers a solution.A benevolent aid officer may still be tempted to always split the aid more or less equallyamong the countries. Our paper suggests that holding a contest among recipient countriescan overcome this problem. That is, instead of giving the aid officer full discretion over thetotal budget for multiple countries, the budget could be partitioned into fixed “prizes” thatthe officer allocates to the countries. Obviously, it may be politically difficult to implementa contest where a country receives zero aid even if it invested effort in reforms. However,some variant of our mechanism, where all countries receive aid but some countries receive“bonus aid” through a contest might be both politically feasible and desirable from theincentive point of view.

12Quote of a management adviser in “For Whom the Bell Curve Tolls”, J. McGregor, The WashingtonPost, November 20, 2013.

13In “‘Rank and Yank’ Retains Vocal Fans”, L. Kwoh, The Wall Street Journal, January 31, 2012.

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The paper is organized as follows. Section 2 describes the model. In Section 3 weshow that the set of optimal contracts contains a contest, we characterize all optimalcontests, and we discuss contest success functions that implement the optimal outcome. InSection 4 we develop several extensions of the baseline model. Section 5 contains a moredetailed discussion of the related literature. Section 6 concludes with a discussion of severaldisadvantages of contests that our model abstracts from (e.g. sabotage and collusion). Theproofs of our main results are relegated to Appendix A. The proofs of additional resultsare in Appendix B.

2 The Model

2.1 Environment

A principal contracts with a set of agents I = {1, ..., n} where n ≥ 2. Each agent i ∈ Ichooses an effort level ei ≥ 0 and obtains a monetary transfer ti ≥ 0. The agents have anoutside option of zero. The payoff of agent i is given by

πi(ei, ti) = u(ti)− c(ei).

The utility function u : R+ → R is twice differentiable, strictly increasing, weakly concave,and satisfies u(0) = 0. The cost function c : R+ → R is twice differentiable, strictlyincreasing, strictly convex, and satisfies c(0) = 0, c′(0) = 0, and limei→∞ c

′(ei) = ∞. Theassumption of additive separability of transfers and efforts is standard in contract theory,mechanism design, and contest theory. We will discuss the robustness of our results withrespect to non-separable preferences in Section 4.

We denote effort profiles by e = (e1, ..., en) ∈ E and transfer profiles by t = (t1, ..., tn) ∈T . We assume that E = Rn

+ and T = {t ∈ Rn+ |

∑ni=1 ti ≤ T}, where T > 0 can be

arbitrarily large. Our results hold no matter whether T is eventually binding or not. Thepayoff of the principal from an allocation (e, t) is

πP (e, t) = z(e)−n∑i=1

ti,

where z : E → R+ is interpreted as the production function that converts efforts intooutput. For clarity of exposition we will focus only on the case where z(e) =

∑ni=1 ei. Our

main results continue to hold if we assume more generally that z is symmetric, weaklyconcave, and strictly increasing in each of its arguments.

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Example. We will use a parameterized example to illustrate our results throughout thepaper. In this example, each of the n agents has the payoff function

πi(ei, ti) = tαi − γeβi ,

where 0 < α ≤ 1 parameterizes risk-aversion, β > 1 describes the degree of cost convexity,and γ > 0 determines the relative weight of effort costs.14 We will always assume that Tis large enough to be non-binding in the example. The first-best effort level eFB is whatthe principal would demand from each agent if she could perfectly control effort and wouldonly have to compensate the agent for his cost, thus paying tFB = u−1(c(eFB)). In ourexample, maximization of e− u−1(c(e)) yields

eFB =

(α

βγ1/α

) αβ−α

and tFB =

(α

βγ1/β

) ββ−α

.

The principal’s first-best profit is n(eFB − tFB). �

Like in the standard delegation model, the state of nature – the effort profile of theagents – is neither verifiable to outside parties (e.g. a court) nor observable to the principal.The efforts can, however, be observed by a reviewer. Consequently, the evaluation of theagents’ performance and the decision on how to reward the agents are delegated to thereviewer. The payoff of the reviewer from an allocation (e, t) is given by

πR(e, t, θ) = πP (e, t) + θn∑i=1

πi(e, t),

where θ is a parameter that captures how much the reviewer cares about the well-beingof the agents, and thus by how much the reviewer’s preferences are misaligned with thoseof the principal. The parameter θ can be thought of as a fundamental preference or asa reduced-form representation of concerns due to other interactions with the agents. Weassume that θ is private information of the reviewer, observable neither to the principal norto the agents. It is drawn according to a commonly known continuous distribution withfull support on Θ = [ θ, θ ], where θ < θ. We describe this distribution by an (absolutelycontinuous) probability measure τ over Θ. Our results will be independent of the shapeas well as the location of this distribution. In particular, τ could be arbitrarily close to aprobability measure with atoms, and our results would also continue to hold if the supportΘ was a union of countably many disjoint intervals.

In line with the literature on delegation, we assume that the principal does not pay the

14Since we assume α > 0, the example includes some but not all CRRA utility functions. This is becausewe require u(0) = 0, which is not satisfied by all CRRA functions.

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reviewer conditional on the decision that the reviewer makes (i.e., whether he submits apositive or a negative review of the agents). The principal could pay a fixed fee, which wenormalize to zero. In addition, our model can be easily reformulated to include paymentsto the reviewer based on the profitability of the firm, such as when the reviewer owns sharesor has career concerns that align his interests with the performance of the firm. Any suchterm in the reviewer’s payoff function can be captured by πP and a normalization of θ.

2.2 Implementation with Credible Contracts

The timing is as follows. First, the principal delegates the evaluation and remunerationof the agents to the reviewer by designing a set D of possible actions. An action is aprobability measure µ ∈ ∆T on the set of transfer profiles, describing the potentiallystochastic payments made to the agents. Given the delegation set, the agents choose theirefforts simultaneously. The reviewer then observes the efforts and his type and chooses anaction from D to reward or punish the agents.15

Since e and θ are observable only to the reviewer, he is always free to choose any actionthat he prefers. We model this by defining a contract Φ = (µe,θ)(e,θ)∈E×Θ as a collection ofprobability measures µe,θ ∈ ∆T , one for each (e, θ) ∈ E×Θ. The interpretation is that theprincipal suggests that a reviewer of type θ should reward an effort profile e by transfersaccording to µe,θ.16 The following incentive constraint makes sure that the reviewer indeedhas an incentive to follow this suggestion:

ΠR(e, µe,θ, θ) ≥ ΠR(e, µe′,θ′ , θ) ∀(e, θ), (e′, θ′) ∈ E ×Θ, (IC-R)

where

ΠR(e, µe′,θ′ , θ) = Eµe′,θ′

[πP (e, t) + θ

n∑i=1

πi(ei, ti)

].

We say that a contract Φ is credible if it satisfies (IC-R). Given a credible contract, the

15A different timing would be that the reviewer observes his type already before he interacts with theagents and observes their efforts. In that case, the principal could offer a menu of delegation sets fromwhich the reviewer selects one before observing the agents’ efforts. Our results are robust to this differenttiming provided that the reviewer’s selection of a delegation set is not observable to the agents. If it wasobservable, additional signalling issues would arise, because the agents may coordinate on different effortprofiles contingent on the reviewer’s observable action. We refrain from studying these issues here, but weremark that, even in that case, the additional gain (if any) from a more complicated mechanism is limitedif the agents’ risk-aversion is moderate or the number of agents is large (see Section 4).

16This formulation does not preclude the possibility that a reviewer randomizes over actions, becausethe randomization over probability measures can instead be written as a compound measure that is chosenwith probability one. We impose the following regularity condition on contracts: for each measurable setA ⊆ T , µe,θ(A) is a measurable function of (e, θ). This ensures that expected payoffs are well-defined incontracts.

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delegation set is implicitly given by D = {µ ∈ ∆T | ∃(e, θ) s.t. µ = µe,θ}.Denote by σi ∈ ∆R+ agent i’s mixed strategy for his effort provision. We also write

ei ∈ ∆R+ for Dirac measures that represent pure strategies. Strategy profiles are given byσ = (σ1, ..., σn) ∈ (∆R+)n. We also use σ to denote the induced product measure in ∆E.We say that a contract Φ implements a strategy profile σ if it is credible and satisfies

Πi((σi, σ−i),Φ) ≥ Πi((σ′i, σ−i),Φ) ∀σ′i ∈ ∆R+, ∀i ∈ I, (IC-A)

where

Πi(σ,Φ) = Eσ[Eτ[Eµe,θ [u(ti)]

]]− Eσi [c(ei)] .

Since a deviation to an effort of zero always guarantees each agent a payoff of at least zero,the agents’ participation constraints can henceforth be ignored.17

The principal maximizes her expected payoff by choosing a contract Φ to implementsome strategy profile σ. Formally, the principal’s problem is given by

max(σ,Φ)

ΠP (σ,Φ) s.t. (IC-R), (IC-A), (P)

where

ΠP (σ,Φ) = Eσ

[n∑i=1

ei

]− Eσ

[Eτ

[Eµe,θ

[n∑i=1

ti

]]].

A contract Φ∗ is optimal if there exists σ∗ such that (σ∗,Φ∗) solves (P).Finally, we introduce a specific class of contracts that will be referred to as contests. A

contest is described by a collection of prizes and a contest success function. Put differently,a contest commits to a profile of prizes y = (y1, ..., yn) ∈ T , some of which could be zero,and specifies how these prizes are allocated to the n agents as a function of their efforts.More formally, let P (y) denote the set of permutations of y.18 Then a contest Cy withprize profile y is a contract that satisfies (i) µe,θ(P (y)) = 1 for all (e, θ) ∈ E ×Θ, and (ii)

µe,θ = µe,θ′ for all θ, θ′ ∈ Θ and e ∈ E. This formulation includes all common contests,

such as all-pay auctions and Tullock contests.Note that every contest is credible. This is because, once the agents’ efforts are sunk,

any permutation of the prizes generates the same payoff for the principal and the same

17Formally, constraints (IC-R) and (IC-A) characterize Perfect Bayesian equilibria of the following game.Given a delegation set D, the agents first simultaneously choose their efforts e. Nature then determines thereviewer’s type θ. The reviewer finally observes e and θ and chooses from D. Constraint (IC-R) prescribessequential rationality for the reviewer’s (singleton) information sets, while (IC-A) prescribes sequentialrationality (with weakly consistent beliefs) for the information sets in which the agents choose.

18Profile t is a permutation of y if there exists a bijective mapping s : I → I such that ti = ys(i) ∀i ∈ I.

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sum of utilities for the agents. Formally, in any contest Cy we obtain that

ΠR(e, µe′,θ′ , θ) =

n∑i=1

ei −n∑i=1

yi + θ

(n∑i=1

u(yi)−n∑i=1

c(ei)

)

is independent of (e′, θ′). However, the set of credible contracts is substantially larger thanthe set of contests. For instance, it is possible to select from a much larger set of transferprofiles, not just permutations of a given prize profile, and still keep both the expectedsum of transfers and the expected sum of the agents’ utilities constant (see Section 4 foran example). Furthermore, credible contracts do not have to be independent of θ but canscreen different reviewer types.

3 Optimal Contracts

3.1 The Optimality of Contests

To illustrate the key incentive problem in our model, suppose first that the preferenceparameter θ was known to the principal and the agents. The following example showsthat, in this case, there may exist a credible contract which is not a contest but whichimplements the first-best effort levels and extracts the entire surplus.

Example. Consider our previous example for the special case of n = 2. Suppose thereviewer’s type was common knowledge. First assume θ = 0, so that there is also nomisalignment of preferences between the principal and the reviewer. Consider a contractΦFB where, if both agents exert eFB, each of them is paid tFB. If one agent deviates,that agent is paid 0 while the non-deviating agent is paid 2tFB. In case both agentsdeviate, they are again each paid tFB. It is easy to verify that this contract is credible,because the sum of transfers is constant across (tFB, tFB), (2tFB, 0) and (0, 2tFB), whichmakes the reviewer indifferent between these transfer profiles. It is also easy to verifythat this contract implements (eFB, eFB), because both agents receive a payoff of zero inequilibrium and a payoff of at most zero after any unilateral deviation. Thus the first-bestis achievable. Observe that ΦFB is not a contest, because the three transfer profiles arenot all permutations of each other. We will show in Section 3.2 that the first-best is notachievable by a contest if the agents are risk-averse. Hence, ΦFB performs strictly betterthan any contest in this example. This shows that non-verifiability of effort alone does notmake contests optimal.19

19This argument is similar to the optimality of bonus pools in Rajan and Reichelstein (2006). It isalso related to MacLeod (2003), who considers transfers to a third party in order to keep the principal’sexpenditure constant. Fuchs (2015) shows that there is an additional commitment advantage of bonuspayment schemes – they allow the manager to credibly reveal the fit between the worker and the firm.

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Now assume that θ > 0 and adjust the contract ΦFB as follows. The payment 2tFB toa non-deviating agent is replaced by some tnd, while everything else is kept unchanged. Iftnd is chosen such that the reviewer is indifferent between the transfer profiles (tFB, tFB),(tnd, 0) and (0, tnd), credibility is restored and the first-best can be implemented. Forinstance, with α = 1/2, β = 2, and γ = 1 we have eFB ≈ 0.63 and tFB ≈ 0.16. For areviewer of known type θ = 3 we would then obtain tnd ≈ 1.15.20 This shows that themisalignment of preferences per se does not make contests optimal either. �

The contracts described in the example no longer work if θ is the reviewer’s privateinformation. Just consider the optimal contract for type θ = 0. Any reviewer with typeθ′ > 0 will strictly prefer to allocate (tFB, tFB), no matter what efforts the agents haveexerted. This is the leniency bias. Similarly, any reviewer with type θ′′ < 0 would neverallocate (tFB, tFB). The crucial point is that only the reviewer of type θ = 0 will follow thecontract. All other reviewer types, no matter how close they are to type θ = 0, will nottake the efforts of the agents into account when allocating rewards. Thus, the contractsoutlined in the example above are knife-edge cases. They fail as soon as the reviewer’stype is drawn from an arbitrary continuous distribution. This creates a stark contrastbetween the case of perfect information and the case of arbitrarily small uncertainty aboutthe reviewer’s type.

Our first main result shows that, despite the fact that the set of possible contracts isvery large, optimal contracts with uncertainty about θ take a very simple form.

Theorem 1 The set of optimal contracts contains a contest.

We prove Theorem 1 by a series of six lemmas. Since we have shown that the principalmay be able to implement the first-best if she knew the reviewer’s private type θ, it wouldseem reasonable to expect that the principal could benefit from screening these types. Infact, it is not difficult to construct contracts that screen the reviewer’s type, by varying thesum of transfers and the sum of utilities given to the agents. However, Lemmas 1 - 3 belowdemonstrate that it is not possible for the principal to benefit from screening. Lemmas 4- 5 then show that the principal cannot benefit from implementing mixed or asymmetriceffort profiles. Finally, Lemma 6 shows that using contests is without loss of optimality.

To begin with, we fix an arbitrary contract Φ = (µe,θ)(e,θ)∈E×Θ and denote

St(e, θ) = Eµe,θ

[n∑i=1

ti

], Su(e, θ) = Eµe,θ

[n∑i=1

u(ti)

]

20The indifference condition is −2tFB + θ2u(tFB) = −tnd + θu(tnd). Given our parameters, it has asecond solution tnd ≈ 3.72, which would work as well. Note that the indifference condition is not guaranteedto have a solution for all parameter values, so our simple construction of a first-best contract does notwork for all values of θ > 0.

13

and

S(e, θ) = −St(e, θ) + θSu(e, θ).

We can then rewrite the credibility constraint (IC-R) as

−St(e, θ) + θSu(e, θ) ≥ −St(e′, θ′) + θSu(e′, θ′) ∀(e, θ), (e′, θ′) ∈ E ×Θ.

Our first lemma provides a characterization of this multidimensional constraint.

Lemma 1 A contract Φ is credible if and only if the conditions (i) - (iii) hold:

(i) ∀θ ∈ Θ, S(e, θ) = S(e′, θ) ∀e, e′ ∈ E.

(ii) ∀e ∈ E, Su(e, θ) is non-decreasing in θ.

(iii) ∀e ∈ E, S(e, θ) = S(e, θ) +∫ θθSu(e, s)ds ∀θ ∈ Θ.

Conditions (ii) and (iii) are familiar from the mechanism design literature. They haveto hold separately for each fixed effort profile e. Condition (i) concerns the effort dimensionand shows that the payoff of any reviewer has to be constant for any reported e.

Observe that combining (i) and (iii) implies∫ θθSu(e, s)ds =

∫ θθSu(e

′, s)ds for all e, e′

and θ. This means that not only S(e, θ) but also its constituent Su(e, θ) has to be constantacross different effort profiles e (for almost every θ). But then the other constituent St(e, θ)cannot vary with e either. The next lemma summarizes this important implication of thecredibility constraint.

Lemma 2 A contract Φ is credible only if there exists a pair of functions x : Θ→ R+ andx : Θ→ R+ such that, ∀e ∈ E,

St(e, θ) = x(θ), Su(e, θ) = x(θ)

for almost all θ ∈ Θ.

We now show that there is no gain for the principal from screening the reviewer’s privatetype by using a complex contract where µe,θ varies with θ. Put differently, the principalcan without loss of generality design the delegation set in a way such that all reviewersselect the same actions.

Lemma 3 For every contract Φ that implements a strategy profile σ, there exists a contractΦ that also implements σ, yields the same expected payoff to the principal, and, ∀θ, θ′ ∈ Θ,satisfies µe,θ = µe,θ

′ ∀e ∈ E.

14

The proof of the lemma is constructive and shows how the contract Φ without screeningcan be obtained from an arbitrary credible contract Φ. Additional intuition can be obtainedby looking again at the example from the beginning of this section. There, the lenientreviewer of type θ = 3 is compensated for revealing a shirking agent by a larger sum oftransfers that he can allocate to the non-shirking agents off the equilibrium path. However,by Lemma 2, the sum of transfers St(e, θ) cannot vary with the reported effort profile e,so the principal cannot just compensate the reviewer off the equilibrium path. Screeningwould require that the expected sum of transfers also varies with type θ on the equilibriumpath, which is not beneficial to the principal.

Given this result, we from now on focus without loss of generality on contracts wherethe agents’ transfers depend on their efforts only, which we write as Φ = (µe)e∈E. The nextlemma shows that the principal does not benefit from implementing mixed strategies.

Lemma 4 For every contract Φ that implements a strategy profile σ, there exists a contractΦ that implements the pure-strategy profile e = (e1, ..., en), where ei = Eσi [ei] ∀i ∈ I, andyields the same expected payoff to the principal.

The intuition behind this result is simple: any randomness in transfers that is achievedby mixed strategies can equivalently be generated by the contract. On the other hand,since c is convex, the agents benefit from exerting the average effort ei instead of σi,while the principal is indifferent as to whether she obtains the efforts in expectation ordeterministically.

The next lemma states that, in the current symmetric setting, it is without loss torestrict attention to the implementation of symmetric effort profiles.

Lemma 5 For every contract Φ that implements a pure-strategy profile e = (e1, ..., en),there exists a contract Φ that implements the symmetric pure-strategy profile e = (e1, ..., en),where e1 = . . . = en = 1

n

∑ni=1 ei, and yields the same expected payoff to the principal.

The next lemma completes the proof of Theorem 1 by demonstrating that the principalcan achieve the same payoff with a contest as with any non-screening contract that imple-ments a symmetric pure-strategy effort profile. Thus, the principal can obtain her maximalpayoff with a contest.21

Lemma 6 For every contract Φ that implements a symmetric pure-strategy profile e, thereexists a contest Cy that also implements e and yields the same expected payoff to the prin-cipal.

21To be exact, Theorem 1 follows only after it has been shown that problem (P) has a solution, so thatan optimal contract exists. This will be shown in the proof of Theorem 2 in the next section.

15

To prove this lemma, we construct a contest which implements the effort profile e. Thiscontest features n − 1 identical prizes and one prize that is smaller. The small prize isused to punish agents who deviate in either direction from e. In equilibrium, when theeffort profile e is realized, the small prize is allocated randomly among the agents. As wewill show in the next section, this particular prize structure is a general feature of optimalcontests, while the specific (non-monotonic) allocation rule is not required to achieve theoptimum.

3.2 Optimal Contests

From the previous section, we know that the principal can restrict attention to contestswhen designing an optimal contract. In this section, we characterize general features of alloptimal contests. When describing a contest Cy, in the following we always assume w.l.o.g.that the prize profile y is ordered such that y1 ≥ y2 ≥ . . . ≥ yn.

Theorem 2 A contest is optimal if and only if the conditions (i) and (ii) hold:

(i) The prizes satisfy yn = 0 and∑n

k=1 yk = x∗, with x∗ = min{x, T} and x given by

u′(

x

n− 1

)= c′

(c−1

(n− 1

nu

(x

n− 1

))).

If the agents are risk-averse, then the prize profile is unique and given by

y = (x∗/(n− 1), . . . , x∗/(n− 1), 0).

(ii) The contest implements (e∗, . . . , e∗), where e∗ is given by

e∗ = c−1

(n− 1

nu

(x∗

n− 1

)).

Condition (i) in the theorem shows that the lowest prize will be zero in any optimalcontest. This is not obvious, since the agents can be risk-averse and in equilibrium allagents face the risk of receiving the zero prize. The intuition is that in equilibrium anagent receives the zero prize with probability 1/n, while a shirking agent would receive thezero prize with strictly larger probability (possibly one). Thus, any increase in yn decreasesthe difference between the equilibrium and the deviation payoffs, and therefore decreasesthe amount of effort that can be demanded in equilibrium. When the agents are risk-averse,the optimal prize profile will feature n− 1 identical positive prizes in addition to the zeroprize, which keeps the risk imposed on the agents in equilibrium at a minimum. Thisprize structure is not novel, and its good incentive properties have been observed before

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in different settings.22 We also characterize the optimal total prize sum x∗, which is givenby the point x where marginal cost and benefit of inducing effort are equalized, or by theexogenous budget T whenever it is sufficiently tight.23

Condition (ii) in the theorem shows that every optimal contest extracts the entiresurplus from the agents, because it implements a symmetric pure-strategy effort profilesuch that each agent’s equilibrium payoff is zero. This condition puts limits on the setof possible contest success functions which can be used to achieve the optimum. We willexplore these limits in the following subsection.

Having characterized the optimal contests, we turn to the question of efficiency loss.

Corollary 1 If the agents are risk-neutral, the principal can achieve the first-best. If theagents are risk-averse, the principal cannot achieve the first-best.

The efficiency loss is driven entirely by risk-aversion of the agents. The loss is a directconsequence of the necessity to assign a prize of zero. Since the principal has to committo the low prize, it must be delivered even in equilibrium. Risk-averse agents have to becompensated for this, which increases the cost of inducing effort. Hence, the commitmentproblem prevents the principal from achieving the first-best. However, the loss will besmall if the agents are only mildly risk-averse, as the following example illustrates.

Example. Consider again our example for n = 2. Applying the results from Theorem 2,it can be shown that e∗ = 2

α−1β−α eFB and x∗ = 2

β−1β−α tFB holds in any optimal contest. The

ratio of second-best to first-best profits of the principal is therefore

R =2e∗ − x∗

2eFB − 2tFB=

2β−1β−α (eFB − tFB)

2(eFB − tFB)= 2

α−1β−α .

This ratio is increasing in α, with R → 1 in the limit as α → 1, so second-best profitsapproach first-best profits when the agents’ risk-aversion vanishes. �

22For example, Nalebuff and Stiglitz (1983) show that a single negative prize (a penalty) provides betterincentives than a single positive prize. Glazer and Hassin (1988) show that the optimal way to split a budgetinto prizes is to have n − 1 positive prizes and one prize equal to zero. The reason for optimality of thisprize profile is the same as in our model, namely the assumption that agents are risk-averse. It is difficultto directly compare these papers to ours, since they consider only one specific contest success function thatis taken as given. There are also other differences to our model, e.g. Nalebuff and Stiglitz (1983) assumethat principals always earn zero profits due to perfect competition.

23Given that our optimal prize structure has n− 1 identical prizes, one could contemplate a mechanismin which the reviewer is forced to pay zero to one agent but is otherwise free to allocate x∗ arbitrarilyamong the remaining agents. A reviewer with θ ≥ 0 would find it optimal to hand out n− 1 equal shares.Hence, if θ ≥ 0 is guaranteed, this mechanism is just an “indirect” implementation of the optimal “direct”contest mechanism (in particular, it falls under our formal definition of a contest). This is no longer trueif θ < 0 is possible, in which case the reviewer would find it optimal to pay x∗ to a single agent, destroyingthe optimality of the indirect mechanism.

17

The loss will also be small if there are many agents, because the probability of notreceiving any of the n− 1 identical prizes is 1/n in equilibrium and vanishes as the numberof agents grows. This has been observed before by Glazer and Hassin (1988), and similararguments can be found in Green and Stokey (1983) and Nalebuff and Stiglitz (1983).Here, these arguments imply that delegation comes with little loss compared to an uncon-strained mechanism design approach with monetary transfers if risk-aversion is small or ifthe number of agents is large.

3.3 Unique Implementation

We say that a contract Φ uniquely implements some pure-strategy effort profile e if it(i) implements e and (ii) does not implement any other (possibly mixed) strategy profileσ 6= e. The next theorem states that the second-best effort profile from Theorem 2 can beuniquely implemented by an all-pay auction with a cap.24

An all-pay auction is one of the canonical contest types (see Konrad, 2009, Ch. 2.1).It is perfectly discriminating, in the sense that the agent with the highest effort wins thehighest prize with probability one, the agent with the second highest effort wins the secondprize, and so on. Ties are broken randomly. A cap is a maximum admissible effort level(see Che and Gale, 1998). In our setting, a cap could be implemented in two different ways.The reviewer can be instructed to not differentiate effort levels at or above the cap, i.e.,an agent who exerts effort exactly at the cap and an agent who exerts effort above the capare treated the same and have the same chance of winning each of the prizes. With ouroptimal prize profile, this amounts to the instruction that the low price be allocated to theagent with the lowest effort, except if all agents have reached a pre-specified performancethreshold (the cap), in which case the low prize is allocated randomly. It is a best responseof the reviewer to follow this instruction.

An even simpler implementation is to put an actual upper bound on the effort thateach agent can provide, e.g. by enforcing maximal work hours, limiting the accumulation ofovertime, or imposing page limits and deadlines on grant proposals. This can be done eventhough the principal cannot observe or control the actual effort. For example, by limitingthe work day to eight hours, the principal imposes an upper bound on the maximal effortby limiting the opportunities to exert effort. Making sure that an agent actually exertseffort during those eight hours is a much more difficult problem. Hence, while imposingupper bounds on effort is often feasible, enforcing lower bounds is typically not possible.See Gavious, Moldovanu, and Sela (2002) for many other examples of actual caps that are

24Uniqueness, as defined here, refers to the agents’ choice of efforts in the given contest. The reviewerwill always be indifferent among several actions, by Lemma 1. In particular, a “babbling” equilibrium existsin any credible contract. In such an equilibrium, the reviewer’s behavior is unresponsive to the agents’efforts and they exert zero effort. Hence, our notion of uniqueness is the strongest possible in this setting.

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imposed in different contests.The following result shows that a cap at the optimal effort level e∗ generates a unique

equilibrium in the all-pay auction, which is in pure strategies.

Theorem 3 The effort profile (e∗, ..., e∗) is uniquely implemented by an all-pay auctionwith prize profile y = (x∗/(n− 1), ..., x∗/(n− 1), 0) and a cap at e∗.

To see why all agents exerting e∗ is an equilibrium, observe that upward deviationsare ineffective (or impossible) while downward deviations guarantee the zero prize. Theintuition for the result that no other pure-strategy equilibria exist is similar to that forall-pay auctions without caps: For every effort profile e 6= (e∗, ..., e∗), either an upwarddeviation discretely increases the probability of winning, or a downward deviation decreasescosts without changing the probability of winning. The last step in the proof is to showthat the cap destroys any potential mixed-strategy equilibrium.25

An existing literature has investigated the effect of caps on equilibria and revenues inall-pay auctions (see Che and Gale, 1998, for an early contribution and Olszewski andSiegel, 2017, for a very general recent treatment). In particular, it is known that caps cangenerate pure-strategy equilibria. In our setting, e∗ is the largest possible cap for which apure-strategy equilibrium still exists. This is intuitive. The principal wants to make thecontest as competitive as possible without destroying the pure-strategy equilibrium.

The all-pay auction with a cap is not the only contest success function that implementsthe optimum. It can be shown that a suitably defined Tullock contest is also able to im-plement the optimal effort profile.26 This provides a novel foundation for the imperfectlydiscriminating Tullock contest success function. The noise in the prize allocation is ex-plicitly generated within the optimal contract, rather than being the result of uncontrolledfactors in the allocation process. Combined with the above Theorem 3, this also showsthat the essential feature of our main result is the fixed profile of prizes, and not the exactway in which these prizes are awarded to the agents.

4 Extensions

4.1 Imperfect Effort Observation

In this section, we relax the assumption that the reviewer perfectly observes the effort ofeach agent. We do not want to impose one specific observational constraint, so we firstintroduce a very general information structure. We then derive conditions under which an

25Note that the all-pay auction without a cap does not have an equilibrium in pure strategies. Therefore,the max-min optimal ranking mechanism in Frankel (2014) would not be optimal in our setting withendogenous efforts, because it is unable to implement the optimal effort profile.

26Details are available upon request.

19

optimal contract can be described despite the generality of the framework, and we illustratethe approach with two specific examples.

Suppose that, after the agents have chosen their efforts e = (e1, ..., en) ∈ E, the reviewerobserves a signal s ∈ S that is drawn according to an effort-dependent probability measureηe ∈ ∆S. We denote η = (ηe)e∈E and call (S, η) the observational structure of the model.We do not impose any assumptions on the set of signals S or the stochastic signal-generatingprocess η.27 Hence a large range of applications and examples can be modelled by differentobservational structures. Our previous setting with perfect observation is a special casewhere S = E and ηe is the Dirac measure on e. A second example is the classical moral-hazard setting where each agent’s effort ei produces a random output si such that Eηe [si] =

ei. Only the realized output vector s ∈ S ⊆ Rn becomes observable to the reviewer.It is possible to assume that the principal cares about output rather than effort in thisapplication, because her expected payoffs are unaffected by the noise with zero mean. Athird example is a setting where only an aggregate statistic of the effort profile becomesobservable. For instance, suppose there are two agents and only the difference betweentheir efforts but not the levels can be observed. This amounts to an observational structurewhere S = R and ηe is the Dirac measure on e1− e2. One could also model the observationof ordinal performance ranks, or a blind review process where the individual efforts areanonymized. Finally, the observational structure allows for stochastic signals which arecorrelated across the agents like in Green and Stokey (1983) or Nalebuff and Stiglitz (1983).

Given an observational structure (S, η), a contract is a collection Ψ = (ψs,θ)(s,θ)∈S×Θ,where ψs,θ ∈ ∆T describes the transfers that a reviewer of type θ who has observed signals pays to the agents. As before, a contest Ky with prize profile y is a contract that satisfiesψs,θ(P (y)) = 1 for all (s, θ) ∈ S ×Θ, and ψs,θ = ψs,θ

′ for all θ, θ′ ∈ Θ and s ∈ S.Given a contract, consider a reviewer of type θ who has observed the signal s. This

reviewer will form a posterior belief about the agents’ effort choices, which we denote byςs,θ ∈ ∆E. His payoff when choosing action ψs′,θ′ is

ΠR(s, ψs′,θ′ , θ) = Eςs,θ

[Eψs′,θ′

[πP (e, t) + θ

n∑i=1

πi(ei, ti)

]].

The contract is credible if ΠR(s, ψs′,θ′ , θ) is maximized by ψs′,θ′ = ψs,θ for all (s, θ) ∈ S×Θ.

While the reviewer’s payoff now depends on his endogenous belief ςs,θ, it is easy to see thatthe shape of ςs,θ is irrelevant for the definition of credibility. We can therefore ignore howthis belief is formed in equilibrium. Note also that contests remain credible, because thereviewer’s payoff is independent of (s′, θ′) in any contest.

27Mirroring the remark in footnote 16, we only need the regularity condition that ηe(A) is a measurablefunction of e for each measurable subset A ⊆ S, to ensure that expected payoffs remain well-defined.

20

Given a credible contract, the payoff of agent i with strategy profile σ is

Πi(σ,Ψ) = Eσ[Eηe

[Eτ[Eψs,θ [u(ti)]

]]]− Eσi [c(ei)] .

The contract implements σ if Πi((σi, σ−i),Ψ) ≥ Πi((σ′i, σ−i),Ψ) for all σ′i ∈ ∆R+ and i ∈ I.

Finally, the principal maximizes

ΠP (σ,Ψ) = Eσ

[n∑i=1

ei

]− Eσ

[Eηe

[Eτ

[Eψs,θ

[n∑i=1

ti

]]]]

by choosing a credible contract Ψ and a strategy profile σ to be implemented.As the following result shows, optimal contracts can sometimes be described despite

the generality of this framework.

Proposition 1 Fix an arbitrary observational structure (S, η). A contest Ky with prizeprofile y = (x∗/(n− 1), . . . , x∗/(n− 1), 0) is optimal if it implements (e∗, . . . , e∗).

A contest with a prize profile as characterized in Theorem 2 remains optimal for anarbitrary observational structure, provided that it can still implement the effort profile(e∗, ..., e∗) and therefore achieves the same maximal payoff for the principal as in the case ofperfect observation. This is not an obvious result. It is well-known that coarser informationcan be beneficial for a principal in a setting with a commitment problem (see e.g. Konrad,2001). Here, however, the outcome under a coarser observational structure can always bereplicated by some contract when observation is perfect, by emulating the observationalfriction within the contract. The cap in the all-pay auction (or the noise in a Tullockcontest) serve precisely this goal. Hence, coarse or noisy effort observation of the reviewercannot help the principal, so that her maximal payoff with perfect observation is also anupper bound on her payoff with imperfect observation.

Despite its simplicity, the result is a powerful tool for the design of optimal contracts.Whenever we can find a signal-contingent contest success function that ensures implemen-tation of (e∗, . . . , e∗) with prize profile (x∗/(n − 1), . . . , x∗/(n − 1), 0), we can be sure tohave constructed an optimal contract for the given observational structure. We illustratethe applicability of this tool in two simple examples.

Example. Consider a classical moral-hazard setting with two agents. The effort costfunction is given by c(ei) = γeβi for some γ > 0 and β > 1. The noise in output takes amultiplicative form: the output of agent i who exerts effort ei is given by si = eiri, wherethe pair of random variables (r1, r2) follows a bivariate log-normal distribution,

(r1, r2) ∼ lnN

[(ν1

ν2

),

(σ2

1 σ12

σ12 σ22

)].

21

We show in the appendix that the optimal effort profile (e∗, e∗) can indeed be implementedby a contest with prize profile (x∗, 0) whenever the inequality

σ21 + σ2

2 − 2σ12 ≤ 2/(πβ2) (1)

is satisfied, which just requires that the noise in output is not too strong. The contest thatachieves implementation of (e∗, e∗) – and which is therefore an optimal contract by Proposi-tion 1 – allocates the positive prize x∗ to agent 1 with a probability that is increasing in therealized ratio of outputs s1/s2. More precisely, agent 1 receives the prize whenever s1/s2 islarger than a log-normally distributed random number. Similar contests with multiplicativenoise have been studied in the literature.28 With this construction, the overall randomnessin the prize allocation can be adjusted to a level that guarantees implementation of (e∗, e∗).Note that positive correlation in outputs across agents (σ12 > 0) slackens inequality (1)and thus reinforces the optimality of contests, in line with the results by Green and Stokey(1983) or Nalebuff and Stiglitz (1983).

Example. Consider a setting with two agents in which only the difference s = e1− e2 butnot the entire profile (e1, e2) can be observed. We show in the appendix that, despite thisobservational constraint, the optimal effort profile (e∗, e∗) can always be implemented bya contest with prize profile (x∗, 0). The contest that achieves implementation of (e∗, e∗) –and is therefore optimal – allocates the positive prize to agent 1 with a probability that isincreasing in the observed difference s. More precisely, agent 1 receives the prize whenevers is larger than a uniformly distributed random number. Such contests with additive noisehave also been studied in the literature.29 An appropriate level of randomness in theallocation rule again ensures that unilateral deviations from (e∗, e∗) are not profitable.

Of course, Proposition 1 is not always applicable. For instance, it is clear that only zeroeffort can be implemented if the reviewer’s signals are completely uninformative (ηe = ηe

′

for all e, e′ ∈ E). More generally, it will be impossible to implement the second-best effortprofile (e∗, . . . , e∗) when the signals on which the prize allocation can be conditioned aretoo coarse or too noisy. We leave a characterization of optimal contracts for such third-bestenvironments to future research.

28See, for instance, Jia et al. (2013). We are not aware of an explicit treatment of the multiplicativelog-normal noise model in the literature, but of course it can be transformed into a specific probit modelwith additive normal noise (Dixit, 1987).

29See, for instance, Lazear and Rosen (1981). Che and Gale (2000) provide a general treatment ofcontests with additive uniform noise. They show that these contests often do not have a symmetric pure-strategy equilibrium. The uniform distribution used in our construction is chosen precisely to avoid thisproblem.

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4.2 Non-Separable Utility and Favoritism

The assumption that the agents have additively separable utility functions is a natural start-ing point. It matters mostly for our characterization of the reviewer’s credibility constraint,where separability implies that the agents’ sunk efforts do not influence the reviewer’s pref-erences over the distribution of the prizes. Without separability, different prize allocationsmay lead to different sums of the agents’ utilities, which implies that the reviewer mayno longer be indifferent between all possible permutations. In order to keep comparisonssimple, we therefore introduce non-separability directly into the reviewer’s utility func-tion, leaving the agents’ utility functions unchanged. For instance, non-separability can becaptured by modifying the reviewer’s payoff function to

πR(e, t, θ) = πP (e, t) + θn∑i=1

h(πi(e, t)),

for a strictly concave function h : R → R. Whenever θ > 0, which we will assume for thefollowing discussion, this transformation implies a concern for equality of the entire utilitiesof the agents (“wide bracketing”), rather than just their utilities from monetary transfers(“narrow bracketing”). In particular, the reviewer will prefer giving larger prizes to agentswho have exerted higher efforts.30

For a contest to remain credible, it now has to be perfectly discriminating. This rulesout contest success functions such as the Tullock form. However, as we have shown inSection 3.3, the optimum can be achieved with an all-pay auction, which is perfectly dis-criminating. The only complication arises from the necessity of a cap. If efforts above e∗ arepossible, then a reviewer with non-separable utility may not want to follow the instructionto not differentiate between effort levels at and above the cap. Hence, to ensure credibilityeven with non-separable preferences, the cap has to be implemented as an actual upperbound on efforts. A third way of implementing the cap, which we have not discussed sofar, is by using tools from information design. Suppose the principal can shape the obser-vational structure (S, η) introduced in the previous subsection. She could then structurethe observation process such that deviations above e∗ become unobservable to the reviewer(e.g. by forbidding the manager to call the agents outside working hours) and conduct anotherwise standard all-pay auction. Overall, two of the three possible implementations ofthe cap remain credible with non-separable preferences.

A very similar problem is favoritism. The reviewer may not be indifferent between prizepermutations if she prefers some agents over others. To address this problem, the principal

30See Corchon and Dahm (2011) for a characterization of contest success functions which arise if acontest organizer ex post allocates prizes to maximize some non-separable, non-expected utility function.Similarly, Corchon and Dahm (2010) investigate contest success functions which arise when a contestorganizer of unknown type freely allocates prizes ex post.

23

could structure the observation process such that the chosen efforts are observable but notthe identity of the agent who chose each effort. This is commonly referred to as a blindreviewing process and is indeed used for performance evaluation (for example, see Goldinand Rouse, 2000). Garbling the reviewer’s information in such a way comes at no loss forthe principal in an optimal contest, while it ensures credibility even when the reviewer isbiased towards some agents. If blind reviews are impossible, a contest may still performbetter than alternative contracts. For instance, suppose the reviewer has a single favoriteagent that he prefers over the other agents. Then this agent can be sure to receive a positiveprize in the contest and will not exert any effort. However, the contest remains credibleamong the remaining agents and will elicit positive efforts (though not the second-bestlevels). In contrast, bonus pool contracts, where the reviewer allocates a budget freelyamong the agents, suffer even more from favoritism, as the reviewer would allocate all (or adisproportionate share of) the budget to the favorite agent, destroying incentives for effortprovision also among the non-favorite agents. Similar arguments apply to contracts thatgrant the reviewer even more flexibility, such as piece-rate contracts.

The previous arguments show that contests are (or can be made) robust in terms ofthe outcome that can be implemented. However, it is not clear that this outcome remainsoptimal with non-separable preferences or favoritism by the reviewer. The principal maybe able to exploit the particular preferences of the reviewer by writing a contract whichis not a contest. The answer to this question will depend on the exact nature of theprincipal’s knowledge. With precise knowledge of preferences, it may indeed be the casethat a standard contest is no longer optimal. We leave this extension to future research.However, it is worth pointing out again that a robust contest will often achieve an outcomevery close to the first-best, leaving little additional gain from designing a more complexmechanism. This is the case if the number of agents is sufficiently large (Glazer and Hassin,1988; Green and Stokey, 1983; Nalebuff and Stiglitz, 1983) or if the agents’ risk-aversion ismoderate. We illustrate this in the following example.

Example. Consider our running example and fix β = 2 and γ = 1. Figure 1 depicts thepercentage of first-best payoffs that the principal can achieve with an optimal contest, asa function of the risk-aversion parameter α and for several values of n. Two observationsare immediate. First, as α → 1 the share of the first-best payoffs that the principal cancapture converges to one. Second, for any given level of risk-aversion, the principal obtainsa larger share with a larger number of agents. The example also shows that, even for amodest number of agents, the principal obtains a substantial share of the first-best payoffsby running an optimal contest. For instance, already for n = 6 the principal captures morethan 90% of the first-best payoffs for any α ∈ (0, 1). �

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

α

PercentofFBpayoff

n=8

n=6

n=4

n=2

Figure 1: Share of first-best payoffs with an optimal contest.

4.3 Heterogeneous Abilities

We now consider a variation of the basic model in which the payoff of agent i is given by

πi(ei, ti) = u(ti)− ci(ei),

where the cost functions ci satisfy our previous assumptions but can be different acrossagents. We first show that our main result is fully robust for the case of two agents.

Proposition 2 Suppose n = 2. For any profile of effort cost functions (c1, c2), the set ofoptimal contracts contains a contest.

To understand the logic of the result, first note that our earlier Lemmas 1 to 4 continueto hold under cost heterogeneity for any n. Hence it is still without loss of generality toconsider contracts that do not screen the reviewer’s type, implement a possibly asymmetricpure-strategy effort profile, and distribute a fixed expected sum of transfers x. For any suchcontract, we then show that a contest with prize vector y = (x, 0) can implement the sameeffort profile when there are two agents. In contrast to our previous result for the symmetriccase, the agents’ winning probabilities will typically not be identical in equilibrium, becausethe agents may have to be compensated for different effort costs. The following exampleillustrates this point.

Example. Consider a variant of our running example with n = 2, α = 1, and hetero-geneous effort costs ci(ei) = γie

βi for γ1 < γ2. It can be shown that it is optimal for the

25

principal to implement the (first-best) effort levels

e∗1 =

(1

βγ1

) 1β−1

and e∗2 =

(1

βγ2

) 1β−1

.

This can be achieved by a contest with prize vector y = (x∗, 0) for x∗ = c1(e∗1) + c2(e∗2).In equilibrium, agent i receives the prize x∗ with probability p∗i = ci(e

∗i )/(c1(e∗1) + c2(e∗2)).

Hence the more efficient agent 1 is asked for higher equilibrium effort (e∗1 > e∗2). In thisexample, the resulting difference in equilibrium efforts is so large that agent 1 bears ahigher total cost (c1(e∗1) > c2(e∗2)). To compensate this cost difference, the prize x∗ mustbe allocated to agent 1 with a probability larger than 1/2. �

Generalizing the above argument to the case of n > 2 agents and a contest with n− 1

equal prizes faces the difficulty that some agents may have substantially higher effort costsin equilibrium than others and cannot be compensated even if they win one of the identicalprizes for sure. Our next result rests on the insight that effort profiles for which the agents’costs are so strongly heterogeneous cannot be optimal if the agents’ cost functions are notstrongly heterogeneous. In that case, a contest with n − 1 equal prizes and one prize ofzero is still optimal. To formalize this idea, we fix any sequence of cost function profiles(cm1 , . . . , c

mn )m∈N such that, for each i ∈ I, the sequence (cmi )m∈N converges uniformly to

some common cost function c as m→∞.

Proposition 3 Suppose (cm1 , . . . , cmn ) → (c, . . . , c) uniformly. Then there exists m ∈ N

such that for all m ≥ m the set of optimal contracts contains a contest.

A loser-gets-nothing contest remains optimal whenever the differences in abilities arenot too large. Again, the optimal contest will typically ask for different effort levels fromdifferent agents, and it allocates the zero prize with non-identical probabilities across theagents in equilibrium. For example, suppose that the cost heterogeneity is small enoughfor Proposition 3 to apply. Denote by (e∗1, ..., e

∗n) the optimal effort profile and let e∗ =

mini∈I e∗i . This optimum can be implemented by a modified all-pay auction with a cap.

For any real effort profile e = (e1, . . . , en), we first compute a “virtual effort profile” e =

(e1, . . . , en) by ei = (e∗/e∗i )ei for every agent. Then the prizes are allocated as in an all-payauction with cap e∗ in which e = (e1, . . . , en) is taken as the actual efforts exerted by theagents. In equilibrium, when e = (e∗1, . . . , e

∗n) and thus e = (e∗, . . . , e∗), ties are broken such

that agent i receives one of the positive prizes with probability p∗i = ci(e∗i )/u(x∗/(n− 1)),

which is exactly enough to compensate for his effort cost. Thus the auction handicapsagents who have to provide higher efforts by reducing their performance to lower virtualefforts, and compensates those agents who experience higher equilibrium effort costs bybreaking ties with higher probability in their favor.

26

One may conjecture that a contract with more than two prize levels becomes optimalwhen the agents are very different from each other. This turns out to be true. Our nextresult shows that, for arbitrary heterogeneity in the agents’ effort costs, a generalized contestis always optimal. Such a contest is described by two prize profiles y and yd, where theprizes y are allocated in equilibrium and the prizes yd are allocated after deviations. Inessence, the reviewer gets to choose between two different contests and finds it optimal toselect one of them in equilibrium, while the other one remains an unused (but credible)off-equilibrium threat. The special case of a standard contest arises when y = yd.

Proposition 4 For any profile of effort cost functions (c1, . . . , cn), the set of optimal con-tracts contains a generalized contest.

As the proof of Proposition 4 reveals, the optimal equilibrium prize profile y no longerfeatures n−1 equal prizes when costs are sufficiently heterogeneous. Instead, if there are kdifferent levels of equilibrium effort costs, there will usually be k+1 different prize levels.31

That is, there will be one prize level for each cost type and one “low” prize. In equilibrium,the n− 1 agents with highest cost of effort either receive the prize intended for their costtype or the low prize (with some probability). The agent with the lowest cost of effortreceives with positive probability any of the n prizes. A deviation is always punished witha zero prize, that is, the deviation prize vector yd always includes a zero prize.

While real-world contests often feature only two prize levels (for instance, studentssometimes receive only pass-fail grades, tenure is either granted or declined, and workersare promoted or not), there are also contests with multiple prize levels. GE under JackWelch separated their employees into three performance levels,32 and grades are oftengiven on a scale from A to D. According to our theory, multiple prize levels are offered asa response to heterogeneity in the agents’ abilities. The prediction of our model is thatmultiple prize levels will be more likely in contests which are held among a population ofagents where heterogeneity is high (for example, students in a public school), and thattwo prize levels will be more likely if the population of agents is more homogeneous (forexample, junior analysts in a consulting firm).33

31If all agents’ equilibrium effort costs are different or if the heterogeneity is extreme (defined preciselyin the proof), there will be k different prize levels.

32See “‘Rank and Yank’ Retains Vocal Fans”, L. Kwoh, The Wall Street Journal, January 31, 2012.33There are other reasons for using multiple prize levels. Moldovanu and Sela (2001) find, in a model with

incomplete information and risk-neutral agents, that multiple prizes can be optimal for effort maximizationwhen cost functions are convex. Olszewski and Siegel (2018) develop a novel approach to contest design,which can be used for very general classes of large contests with many agents. They characterize thedistribution of prizes which maximizes the aggregate effort exerted by the agents. Among other results,they find that multiple prizes of different levels are optimal when agents have convex costs or are risk-averse.

27

5 Related Literature

Our contribution is related to three distinct groups of papers. First, the optimal delegationmechanism in our paper is a contest, so we contribute to the literature examining conditionsunder which contests are optimal incentive mechanisms. Second, we work with a three-tiered hierarchical structure with a lenient reviewer. There are several papers featuringa similar structure. Third, the principal in our model delegates the decision on how toreward the agents to the reviewer. Our paper is therefore related to the literature onoptimal delegation. We will discuss the connections and differences of our paper to each ofthese strands of literature in turn.

Optimality of contests. In their seminal paper, Lazear and Rosen (1981) showthat contests can implement the socially optimal effort levels when agents are risk-neutral.They assume perfectly competitive labor markets in which the agents obtain all the surplus.Contests are then among the optimal mechanisms because they can induce first-best effort.At the same time, the set of optimal mechanisms also contains a piece-rate contract, amongothers. For the case of risk-averse agents, Lazear and Rosen (1981) compare the two specificmechanisms of piece-rate contracts and contests. They show that either of them sometimesdominates the other, but they do not establish results on global optimality.

A defining feature of contests is that the payoff of the agents depends on how well theyperform relative to each other. This feature can make contests optimal in the presenceof common uncertainty (see Holmstrom, 1982, and Mookherjee, 1984, for a general suffi-cient statistics approach to relative performance pay). Both Green and Stokey (1983) andNalebuff and Stiglitz (1983) show that contests can do better than individual contractswhen agents are risk-averse and there is a random common shock to their outputs. If therelationship between effort and output is ambiguous and the agents are ambiguity-averse,Kellner (2015) shows that contests can be optimal because they filter out the commonambiguity.

In our paper, contests provide a commitment for lenient reviewers to punish shirkingagents. Contests are optimal because they incentivize the agents at least as efficiently asany other contract that can provide such a commitment.

Lenient reviewers. Several papers have looked at a three-tiered hierarchy with alenient reviewer. Prendergast and Topel (1996) and Giebe and Gürtler (2012) consider asituation where the reviewer is facing a single agent and the principal can write contractswhere the reviewer’s pay is contingent on his behavior. In Prendergast and Topel (1996),both the reviewer and the principal receive a signal about the worker’s effort. Their mainresult is that leniency need not be costly for the firm, because it can charge the reviewer forexercising leniency. In Giebe and Gürtler (2012), the principal offers a menu of contracts

28

to screen lenient and non-lenient reviewers. They show that if the non-lenient type iscommon enough, the optimal solution can be to pay a flat wage to the reviewer, and relyon the non-lenient type for punishment of agents who shirk. The main difference betweenthese papers and ours is that we consider multiple agents and do not allow contracts whichcondition payment to the reviewer on the reported evaluation.

Svensson (2003) applies a model with a lenient reviewer to the design of allocationmechanisms for foreign aid. The principal wants to use aid to incentivize countries toimplement reforms. In his model, the principal determines the allocation mechanism, butthe aid is then distributed by a country manager whose utility takes into account thewell-being of the target countries. Svensson (2003) proposes a mechanism where eachcountry manager is given a budget for several similar countries but has discretion in howto allocate the aid across countries. He shows that under certain conditions this mechanismcan incentivize countries to reform. Like in our paper, Svensson (2003) considers multipleagents and does not allow for conditional monetary payments to the reviewer. The maindifference is that we solve for the optimal delegation mechanism while Svensson (2003)compares two specific mechanisms.

Optimal delegation. In the usual delegation problem, the agent (the reviewer in oursetting) is better informed about some exogenous state of nature. The principal delegatesa unidimensional decision to the agent, but restricts the set of actions that the agent canchoose. The question is how this set should be designed if the preferences of the principaland the agent are misaligned. The first to formulate the problem and show the existence ofa solution was Holmström (1977, 1984), who focussed on interval delegation sets. Melumadand Shibano (1991) show that the optimal delegation set does not necessarily take the formof an interval. Alonso and Matouschek (2008) and Amador and Bagwell (2013) characterizethe optimal delegation sets in progressively more general environments and find conditionsunder which the optimal delegation set is indeed an interval.

The canonical delegation model has been applied and extended in a number of ways.34

In the multidimensional delegation model by Frankel (2014), optimal mechanisms exhibitwhat he calls the “aligned delegation” property, which means that all agents behave inthe same way as the principal would behave. Our optimal mechanisms also satisfy thealigned delegation property, i.e., the equilibrium behavior of reviewers is independent oftheir type. Krähmer and Kováč (2016) assume that the agent has a privately known typewhich encodes his ability to interpret the private information he receives later on. Theyalso find that screening is not beneficial in a large range of cases. Tanner (2014) obtains ano-screening result in a standard delegation model with an uncertain bias of the agent.

34See Armstrong and Vickers (2010) for an application to merger policy, Pei (2015a) for a model wheredelegation is used to conceal the principal’s private type, Guo (2016) for a model of delegation of experi-mentation, and Frankel (2017) for a model of delegated hiring decisions.

29

Most papers cited above focus on deterministic delegation mechanisms. Kováč andMylovanov (2009) and Goltsman, Hörner, Pavlov, and Squintani (2009) allow for stochas-tic delegation mechanisms and derive conditions under which the optimal mechanism isdeterministic. We allow for stochastic mechanisms and our optimal mechanism is indeednon-deterministic, but in a special sense – the contest prizes are allocated according to aprobabilistic contest success function of the agents’ efforts.

Finally, instead of delegating the decision to the agent, the principal could ask theagent to report the state of nature and take the action herself. This is the frameworkanalysed in the cheap talk literature in the tradition of Crawford and Sobel (1982). SeeChakraborty and Harbaugh (2007) for a model of multidimensional cheap talk in whichranking statements are credible in equilibrium. Several papers ask if the principal is betteroff delegating the decision or just asking for advice. Bester and Krähmer (2008) find that,if the agent needs to exert effort after selecting a project, delegation of the project selectionis less likely to be optimal. Kolotilin, Li, and Li (2013) consider a model of cheap talk wherethe principal can ex ante commit not to take a certain action ex post. Again, they showthat cheap talk with commitment can outperform delegation. On the other hand, Dessein(2002), Krishna and Morgan (2008), and Ivanov (2010) find that, in general, delegation isbetter than cheap talk.35 These issues do not arise in our model. It is easy to see that anoptimal contest from our delegation setting is also an optimal contract in the cheap talksetting, provided the principal can commit to constraining her own actions in the same wayas she can constrain the actions of the reviewer (like in Kolotilin et al., 2013). On the onehand, cheap talk imposes an additional credibility constraint on the principal’s problem,because she must have an incentive to follow the advice of the reviewer. On the other hand,this constraint is always met by a contest, where any allocation of the prizes leads to thesame total expenditure for the principal.

The main difference between our paper and the above discussed literature is that thestate of nature, on which the expert has private information, is exogenous in the delegationliterature, while it is endogenous in our paper. In our model, the reviewer observes theefforts exerted by the agents. Since the incentives of the agents depend on the behaviorof the reviewer, which in turn depends on the given delegation set, the state of nature isaffected by the principal’s choice of the delegation set.

35Fehr, Herz, and Wilkening (2013) and Bartling, Fehr, and Herz (2014) provide experimental evidenceshowing that individuals value decision rights intrinsically, which implies that delegation may not takeplace even when it is beneficial.

30

6 Conclusion

In this paper we have analyzed a three-tiered structure consisting of a principal, a reviewer,and n agents. The principal designs a reward scheme in order to induce the agents to exerteffort. However, the principal does not observe the efforts, so she delegates the allocationof rewards to the reviewer. The reviewer has private information about the utility weightshe puts on the payoffs of the principal and of the agents.

Our main result is that a very simple mechanism, a contest, is optimal. By providinga commitment for the reviewer to punish shirking agents, a contest efficiently incentivizesthe agents to exert costly effort. We also characterize the set of all optimal contests andshow that they have a flat reward structure with n− 1 equal positive prizes and one zeroprize. Finally, we show that the optimum can be uniquely implemented by an all-payauction with a cap. Our results are robust in a range of extensions, including imperfectobservation of efforts and heterogeneous abilities of the agents.

Of course, contests are not flawless mechanisms. Our model ignores two potential draw-backs – sabotage and collusion. First, since winning a contest only requires outperformingyour competitors, an agent may try to sabotage the effort of his competitors instead ofexerting productive effort.36 Whether this is an issue will depend on how difficult it is tosabotage the competitors, and how well the reviewer can observe attempts at sabotage.In our setting, it is credible for the reviewer to assign the zero prize after observing anysabotage attempt. Hence, if sabotage is observable with a sufficiently high probability, itcan be dealt with just like deviations from the optimal effort profile. Second, while oneattractive aspect of contests is that they filter out common productivity shocks, this featurealso makes collusion an attractive option for contestants.37 The extent to which collusiondiminishes incentives in a contest will depend on specific details of the application. AsBandiera et al. (2005) point out, the ability of co-workers to monitor each other’s outputis necessary for collusion. Furthermore, in our all-pay auction with a cap, collusion on aneffort level lower than the cap would be extremely fragile, as a marginal increase in effortwould guarantee the deviator a positive prize. Despite these potential drawbacks, contestsare widely used in reality. Understanding precisely when the benefits of contests outweightheir shortcomings is a promising area for future research.

Other interesting questions could be asked in the framework of delegated performanceevaluation. Here we will mention three immediate ones. First, the framework developedhere can be used to study other forms of reviewer bias. A particularly important bias

36Berger et al. (2013) find that, when given the opportunity, subjects in their laboratory experiment doindeed engage in some sabotage.

37Using personnel data from a large UK farm, Bandiera, Barankay, and Rasul (2005) show that workerspaid under a relative incentive scheme do collude. Fairburn and Malcomson (2001) consider collusion (withside payments) between workers and the supervisor and show how it can reduce incentives for agents toexert effort.

31

that could be examined is the gender bias (Goldin and Rouse, 2000), or discriminationby the reviewer more generally. As we have discussed earlier, a common response to thisproblem is the use of a blind reviewing process. It would be interesting to examine howdifferent biases like leniency and gender bias interact in shaping optimal observationalstructures and contracts. Second, in addition to incentivizing agents, principals will oftenbe interested in screening the abilities of heterogeneous agents in order to be able to assignmore responsibilities to more capable agents. The purpose of a tenure or promotion contestis obviously not only to induce hard work, but also to select the right agents for a moreadvanced position. A question that could be asked in this framework is how screening andprovision of incentives interact when both are delegated to potentially biased reviewers.Third, the principal also hires the reviewer. At first blush, our results might seem tosuggest that the principal would be better off by trying to recruit a selfish reviewer whowill not take the well-being of the agents into account. However, that would be the caseonly if the principal were able to determine the reviewer’s type with absolute certainty.If there is a small remaining uncertainty, our results hold and assert that the principal’smaximal payoff in an optimal contest does not depend on the reviewer’s type. This impliesthat the principal is free to select the reviewer based on other criteria. For instance, analtruistic mid-level manager may outperform a selfish one in uniting his team to face acommon challenge.

In sum, our results offer a novel explanation for the optimality of contests and alsopoint to new applications where contest-like mechanisms could be implemented. Settingswhere an intermediary allocates monetary rewards are widespread in the economy, and, asthe example of foreign aid illustrates, they can be found in unexpected places.

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36

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37

A Proofs of Main Results

A.1 Proof of Theorem 1

A.1.1 Proof of Lemma 1

If-statement. We first show that (IC-R) is implied by (i) - (iii). Note that (IC-R) can berewritten as

S(e, θ)− S(e′, θ′) ≥ (θ − θ′)Su(e′, θ′) ∀(e, θ), (e′, θ′) ∈ E ×Θ.

Using (i) and (iii), this is equivalent to the requirement that, ∀e′ ∈ E and ∀θ, θ′ ∈ Θ,∫ θ

θ′(Su(e

′, s)− Su(e′, θ′)) ds ≥ 0,

and this inequality indeed holds since Su(e′, θ) is non-decreasing in θ by (ii).Only-if-statement. We now proceed to prove that (IC-R) implies (i) - (iii). Note that

for the special case θ′ = θ, (IC-R) is reduced to the requirement that S(e, θ) ≥ S(e′, θ)

∀e, e′ ∈ E. Interchanging e and e′, we immediately obtain (i). Next, consider the specialcase where e′ = e. For this case, (IC-R) requires that, ∀θ, θ′ ∈ Θ,

S(e, θ) ≥ −St(e, θ′) + θSu(e, θ′) (ICθ,θ′)

and

S(e, θ′) ≥ −St(e, θ) + θ′Su(e, θ). (ICθ′,θ)

Summing up (ICθ,θ′) and (ICθ′,θ) we obtain

(θ − θ′) (Su(e, θ)− Su(e, θ′)) ≥ 0 ∀θ, θ′ ∈ Θ.

Thus, Su(e, θ) must be non-decreasing in θ, which is condition (ii). The envelope formulain (iii) follows directly from Theorem 2 of Milgrom and Segal (2002), where absolutecontinuity of S(e, θ) holds because the set of transfer profiles is bounded. �

A.1.2 Proof of Lemma 2

By Lemma 1, credibility implies that, ∀e, e′ ∈ E and ∀θ ∈ Θ,

δ(e, e′, θ) = S(e, θ)− S(e′, θ) =

∫ θ

θ

(Su(e, s)− Su(e′, s)) ds = 0.

38

This implies that, for any fixed e, e′ ∈ E, Su(e, θ) = Su(e′, θ) for almost every θ ∈ Θ. It

then also immediately follows that St(e, θ) = St(e′, θ) for almost every θ ∈ Θ. Now choose

an arbitrary e′ ∈ E and define the functions x and x by

x(θ) = St(e′, θ), x(θ) = Su(e

′, θ) ∀θ ∈ Θ.

It follows that, for any e ∈ E, St(e, θ) = x(θ) and Su(e, θ) = x(θ) for almost all θ ∈ Θ. �

A.1.3 Proof of Lemma 3

Suppose Φ = (µe,θ)(e,θ)∈E×Θ implements σ. In particular, Φ is credible, so by Lemma 2there exists a pair of function x and x such that, ∀e ∈ E,

Eµe,θ

[n∑i=1

ti

]= x(θ), Eµe,θ

[n∑i=1

u(ti)

]= x(θ),

for almost every θ ∈ Θ. Since Φ implements σ, we also have ∀i ∈ I and ∀σ′i ∈ ∆R+,

Eσ[Eτ[Eµe,θ [u(ti)]

]− c(ei)

]≥ E(σ′i,σ−i)

[Eτ[Eµe,θ [u(ti)]

]− c(ei)

].

The expected payoff of the principal with (σ,Φ) is given by

ΠP (σ,Φ) = Eσ

[n∑i=1

ei − Eτ

[Eµe,θ

[n∑i=1

ti

]]]= Eσ

[n∑i=1

ei

]− Eτ [x(θ)] .

For every e ∈ E, define a probability measure µe ∈ ∆T such that

µe(A) = Eτ[µe,θ(A)

]for all measurable subsets A ⊆ T .38 Now construct an alternative contract Φ by settingµe,θ = µe for all (e, θ) ∈ E × Θ. This contract satisfies the property of θ-independencestated in the lemma. Since, ∀(e, θ) ∈ E ×Θ,

St(e, θ) = Eµe,θ

[n∑i=1

ti

]= Eµe

[n∑i=1

ti

]= Eτ

[Eµe,θ

[n∑i=1

ti

]]= Eτ [x(θ)] ,

Su(e, θ) = Eµe,θ

[n∑i=1

u(ti)

]= Eµe

[n∑i=1

u(ti)

]= Eτ

[Eµe,θ

[n∑i=1

u(ti)

]]= Eτ [x(θ)] ,

38The assumption discussed in footnote 16 ensures that the expectation (as well as the ones in the proofof the next lemma) is well-defined. It is also easy to show that µe is indeed a probability measure.

39

by Lemma 1 it is straightforward to check that Φ is credible. Furthermore, note that

Πi(σ′, Φ) = Eσ′

[Eτ[Eµe,θ [u(ti)]

]− c(ei)

]= Eσ′ [Eτ [Eµe [u(ti)]]− c(ei)]

= Eσ′[Eτ[Eµe,θ [u(ti)]

]− c(ei)

]= Πi(σ

′,Φ)

for all σ′ and i ∈ I, which implies that Φ implements σ because Φ implements σ.Finally, from the above arguments we also obtain that the principal’s expected payoff

is Eσ [∑n

i=1 ei]− Eτ [x(θ)] with both (σ,Φ) and (σ, Φ). �

A.1.4 Proof of Lemma 4

Suppose Φ = (µe)e∈E implements σ. We first construct a probability measure η ∈ ∆T by

η(A) = Eσ [µe(A)]

for all measurable subsets A ⊆ T . Furthermore, for each i ∈ I we construct a probabilitymeasure η(i) ∈ ∆T by setting

η(i)(A) = Eσ[µ(0,e−i)(A)

]for all measurable subsets A ⊆ T . We now construct an alternative contract Φ = (µe)e∈E

as follows. For e = e, we let µe = η. For any e = (ei, e−i) with ei 6= ei, we let µe = η(i).For all remaining e, we let µe = µe.

We first show that Φ is credible. Since Φ is credible and its transfers are independent ofθ, by Lemma 2 there exist x, x ∈ R+ such that Eµe [

∑ni=1 ti] = x and Eµe [

∑ni=1 u(ti)] = x

for all e ∈ E. First consider µe for e = e. We obtain

Eµe[

n∑i=1

ti

]= Eη

[n∑i=1

ti

]= Eσ

[Eµe

[n∑i=1

ti

]]= Eσ[x] = x

and, by the analogous argument, Eµe [∑n

i=1 u(ti)] = x. Now consider µe for e = (ei, e−i)

with ei 6= ei. We obtain

Eµ(ei,e−i)

[n∑i=1

ti

]= Eη(i)

[n∑i=1

ti

]= Eσ

[Eµ(0,e−i)

[n∑i=1

ti

]]= Eσ[x] = x

and, by the analogous argument, Eµ(ei,e−i) [∑n

i=1 u(ti)] = x. Since Φ and Φ are identical forall other e, we can conclude that Eµe [

∑ni=1 ti] = x and Eµe [

∑ni=1 u(ti)] = x for all e ∈ E.

40

It is then straightforward to check that Φ is credible by using Lemma 1.We next show that, in Φ, for each agent i ∈ I it is a best response to play ei when

the remaining agents are playing e−i, which implies that Φ implements e. This claim holdsbecause, ∀i ∈ I and ∀e′i 6= ei,

Πi(e, Φ) = Eη[u(ti)]− c(ei)

= Eσ[Eµe [u(ti)]]− c (Eσ[ei])

≥ Eσ[Eµe [u(ti)]]− Eσ[c(ei)] (2)

≥ Eσ[Eµ(0,e−i) [u(ti)]

]≥ Eσ

[Eµ(0,e−i) [u(ti)]

]− c(e′i)

= Eη(i) [u(ti)]− c(e′i) = Πi((e′i, e−i), Φ),

where the first inequality follows the convexity of c and the second inequality follows fromthe fact that Φ implements σ.

Finally, from the above arguments we also obtain that the principal’s expected payoffis∑n

i=1 ei − x with both (σ,Φ) and (e, Φ). �

A.1.5 Proof of Lemma 5

Suppose Φ = (µe)e∈E implements e. We now construct an alternative contract Φ = (µe)e∈E

as follows. For e = e, we define µe by generating a profile of prizes t = (t1, ..., tn) accordingto µe and then allocating these prizes randomly and uniformly among the agents. For anye = (ei, e−i) with ei 6= ei, we let µe by given as follows. A number j is drawn uniformlyfrom I and then a profile of prizes t = (t1, ..., tn) is generated according to µ(0,e−j). Thedeviating agent i gets the prize tj and the remaining n−1 prizes are allocated randomly anduniformly among the non-deviating agents. Note that, by construction, this punishmentrule for unilateral deviations does not depend on the identity of the agent being punished.For all remaining e, we let µe = µe.

We first show that Φ is credible. By Lemma 2, credibility and θ-independence of Φ

imply that there exists x, x ∈ R+ such that Eµe [∑n

i=1 ti] = x and Eµe [∑n

i=1 u(ti)] = x forall e ∈ E. Now first consider µe for e = e. We obtain

Eµe

[n∑i=1

ti

]= Eµe

[n∑i=1

ti

]= x,

Eµe

[n∑i=1

u(ti)

]= Eµe

[n∑i=1

u(ti)

]= x.

41

Now consider µe for any e = (ei, e−i) with ei 6= ei. We obtain

Eµ(ei,e−i)

[n∑i=1

ti

]=

n∑j=1

1

nEµ(0,e−j)

[n∑i=1

ti

]=

1

n

n∑j=1

x = x,

Eµ(ei,e−i)

[n∑i=1

u(ti)

]=

n∑j=1

1

nEµ(0,e−j)

[n∑i=1

u(ti)

]=

1

n

n∑j=1

x = x.

Since Φ and Φ are identical for all other e, we can conclude that Eµe [∑n

i=1 ti] = x andEµe [

∑ni=1 u(ti)] = x for all e ∈ E. It is then straightforward to check that Φ is credible by

using Lemma 1.We next show that, in Φ, for each agent i ∈ I it is a best response to play ei when the

remaining agents are playing e−i, which implies that Φ implements e. To prove this claim,note that

Eµe [u(ti)]− c(ei) ≥ Eµ(0,e−i) [u(ti)]

holds for all i ∈ I because Φ implements e. Summing over all i ∈ I and dividing by n

yields

Eµe[

1

n

n∑k=1

u(tk)

]− 1

n

n∑k=1

c(ek) ≥1

n

n∑k=1

Eµ(0,e−k) [u(tk)] .

We now obtain, ∀i ∈ I and ∀ei 6= ei,

Πi(e, Φ) = Eµe [u(ti)]− c (ei)

= Eµe[

1

n

n∑k=1

u(tk)

]− c(ei)

≥ Eµe[

1

n

n∑k=1

u(tk)

]− 1

n

n∑k=1

c(ek)

≥ 1

n

n∑k=1

Eµ(0,e−k) [u(tk)]− c(ei)

= Eµ(ei,e−i) [u(ti)]− c(ei) = Πi((ei, e−i), Φ),

where the first inequality follows from convexity of c. Hence the claim follows.Finally, from the above arguments we also obtain that the principal’s expected payoff

is∑n

i=1 ei − x with both (e,Φ) and (e, Φ). �

42

A.1.6 Proof of Lemma 6

Suppose Φ = (µe)e∈E implements the symmetric profile e. From the proof of Lemma 5we know that it is without loss of generality to assume that Φ has the following form. Ife = e, a profile of prizes t = (t1, ..., tn) is generated according to some probability measureπ and these prizes are randomly and uniformly allocated to the agents. If e = (ei, e−i) withei 6= ei for some i ∈ I, a profile of prizes td = (td1, ..., t

dn) is generated according to some

(i-independent) probability measure ρ and agent i gets tdn, while the remaining n−1 prizesare randomly and uniformly allocated among the other agents. For all other effort profilese, the transfer rule can be chosen as for e. Thus, we have

Eµe [u(ti)] = Eπ

[1

n

n∑k=1

u(tk)

], Eµ(ei,e−i) [u(ti)] = Eρ[u(tdn)].

Furthermore, by Lemma 2, credibility and θ-independence of Φ imply that there existx, x ∈ R+ such that Eµe [

∑ni=1 ti] = x and Eµe [

∑ni=1 u(ti)] = x for all e ∈ E.

Now construct a contest Cy with prize profile y as follows. Define td as the certaintyequivalent of a deviating agent’s random transfers in contract Φ, i.e., u(td) = Eρ[u(tdn)].Note that td ≤ Eρ[tdn] by concavity of u. Then define the prize profile

y =

(x− td

n− 1, . . . ,

x− td

n− 1, td).

The allocation rule of Cy is as follows. If e = e, the prizes are randomly and uniformlyallocated among all agents. If e = (ei, e−i) with ei 6= ei for some i ∈ I, the deviating agenti obtains td and all other agents obtain (x− td)/(n− 1). For all other effort profiles e, theprizes are again randomly and uniformly allocated among all agents.

Since Cy is a contest, it is credible. Furthermore, ∀i ∈ I and ∀ei 6= ei,

Πi(e, Cy) =

(n− 1

n

)u

(x− td

n− 1

)+

1

nu(td)− c(ei)

≥(n− 1

n

)u

(Eρ[∑n

k=1 tdk)]− Eρ[tdn)]

n− 1

)+

1

nEρ[u(tdn)]− c(ei)

=

(n− 1

n

)u

(Eρ

[n−1∑k=1

1

n− 1tdk

])+

1

nEρ[u(tdn)]− c(ei)

≥(n− 1

n

)Eρ

[u

(n−1∑k=1

1

n− 1tdk

)]+

1

nEρ[u(tdn)]− c(ei)

≥(n− 1

n

)Eρ

[n−1∑k=1

1

n− 1u(tdk)]

+1

nEρ[u(tdn)]− c(ei)

43

= Eρ

[1

n

n−1∑k=1

u(tdk)]

+ Eρ[

1

nu(tdn)]− c(ei)

= Eρ

[1

n

n∑k=1

u(tdk)]− c(ei)

= Eπ

[1

n

n∑k=1

u(tk)

]− c(ei)

≥ Eρ[u(tdn)

]− c(ei)

= u(td)− c(ei) = Πi((ei, e−i), Cy),

where the first inequality follows from td ≤ Eρ[tdn], the second and third inequalities followfrom concavity of u, and the last inequality follows from the fact that Φ implements e. Wecan thus conclude that Cy also implements e.

Finally, from the above arguments we also obtain that the principal’s expected payoffis∑n

i=1 ei − x with both (e,Φ) and (e, Cy). �

A.2 Proof of Theorem 2

Only-if-statement. Suppose (σ∗, C∗y ) solves (P). We first claim that σ∗ must be a pure-strategy effort profile. By contradiction, suppose there exists j ∈ I such that σ∗j is not aDirac measure. We can now proceed exactly as in the proof of Lemma 4 to construct acontract Φ (in fact, a contest) that implements a pure-strategy profile e. The only differenceto the proof of Lemma 4 is that we let ej = Eσ∗j [ej] + ε for some ε > 0 (but still ei = Eσ∗i [ei]for all i 6= j). Credibility of Φ and the fact that ei is a best response to e−i for all i 6= j

follow exactly as in the proof of Lemma 4. Since c is strictly convex and σ∗j is not a Diracmeasure, the first inequality in (2) is strict for j when ε = 0. It then follows that ej is abest response to e−j for sufficiently small ε > 0. Since the principal’s payoff with (e, Φ) isincreased by ε, (σ∗, C∗y ) cannot have been a solution to (P).

Now suppose (e, C∗y ) solves (P), where e may still be asymmetric. Denote x =∑n

k=1 yk.We next show that whenever yn > 0, there exists another contest Cy with

∑nk=1 yk = x

that implements an effort profile e with∑n

i=1 ei >∑n

i=1 ei, and hence (e, C∗y ) cannot havebeen a solution to (P). Denote by pki (e) the probability that agent i receives prize yk in C∗ywhen the effort profile is e. Note that

Πi(e, C∗y ) =

n∑k=1

pki (e)u(yk)− c(ei) ≥n∑k=1

pki (0, e−i)u(yk) ≥ u(yn),

because C∗y implements e. Now consider an agent j ∈ I for which pnj (e) < 1. Constructa contest Cy with a profile of prizes y given by y1 = y1 + δ, yn = yn − δ, and yk = yk for

44

all k 6= 1, n, where δ ∈ (0, yn]. Note that∑n

k=1 yk = x. Let effort profile e be such thatej = ej + ε and ei = ei for all i 6= j, where ε > 0. Note that

∑ni=1 ei >

∑ni=1 ei. The rule of

contest Cy is the following. If the effort profile is e, then the prizes y are allocated such thateach agent i receives prize yk with probability pki (e) = pki (e). If some agent i unilaterallydeviates from e, then agent i receives the prize yn, while the prizes y1, . . . , yn−1 are allocatedrandomly and uniformly among the remaining agents. Otherwise, the allocation of prizescan be chosen arbitrarily. For sufficiently small ε > 0 we then have, ∀i ∈ I and ∀ei ∈ R+,

Πi(e, Cy) =n∑k=1

pki (e)u(yk)− c(ei)

= Πi(e, C∗y ) + p1

i (e)(u(y1 + δ)− u(y1))

+ pni (e)(u(yn − δ)− u(yn)) + c(ei)− c(ei)

≥ u(yn) + p1i (e)(u(y1 + δ)− u(y1))

+ pni (e)(u(yn − δ)− u(yn)) + c(ei)− c(ei)

≥ u(yn − δ) = u(yn) ≥ Πi((ei, e−i), Cy),

where the second inequality holds because

u(yn) + pni (e)(u(yn − δ)− u(yn)) ≥ u(yn − δ)

for all i ∈ I, with strict inequality for j. Hence Cy implements e.When studying the set of all contest solutions to (P), we thus need to consider only

pure-strategy effort profiles e and contests Cy with yn = 0. Fix a sum of prizes x ∈ [0, T ].Let ex be the (unique) effort level that solves

n− 1

nu

(x

n− 1

)− c(ex) = 0.

Note that, by the assumptions on u and c, the solution ex is differentiable, strictly increasingand strictly concave in x. We now claim that nex is an upper bound on the sum ofefforts implementable with a contest Cy that has

∑nk=1 yk = x and yn = 0, and it can be

reached only by implementing the symmetric effort profile (ex, ..., ex). Suppose first thatCy implements an effort profile e with

∑ni=1 ei ≥ nex but e 6= (ex, . . . , ex). Note that

Πi(e, Cy) =n∑k=1

pki (e)u(yk)− c(ei) ≥ u(yn)− c(0) = 0,

45

because Cy implements e. Summing these inequalities over all agents we obtain

n∑i=1

n∑k=1

pki (e)u(yk)−n∑i=1

c(ei) =n−1∑k=1

u(yk)−n∑i=1

c(ei) ≥ 0.

However, due to weak concavity of u and strict convexity of c we also have

n−1∑k=1

u(yk)−n∑i=1

c(ei) < (n− 1)u

(x

n− 1

)− nc(ex) = 0,

a contradiction. Observe next that (ex, . . . , ex) can indeed be implemented. For instance,let y = (x/(n − 1), . . . , x/(n − 1), 0) and choose the rules of Cy as follows. If the effortprofile is (ex, . . . , ex), then the prizes are allocated randomly and uniformly across theagents. If some agent i unilaterally deviates from (ex, . . . , ex), then agent i receives theprize 0, while each other agent receives x/(n− 1). Otherwise, the allocation of prizes canbe chosen arbitrarily. It follows immediately from the definition of ex that this contestindeed implements (ex, . . . , ex).

Given any sum of prizes x, the highest payoff that the principal can achieve is thusgiven by ΠP (x) = nex − x, and the problem is reduced to a choice of x ∈ [0, T ]. Since ΠP

is continuous in x, it follows that a solution exists. Furthermore, since ΠP is differentiableand strictly concave, the first-order condition ∂ΠP/∂x = 0 that is stated in part (i) of thetheorem uniquely characterizes a value x > 0 (given the assumptions on u and c), and theoptimal value of x is given by x∗ = min{x, T}. The resulting implemented optimal effortlevel is then given by e∗ = ex

∗ .We complete the proof of the only-if-statement by showing that any optimal contest has

the profile of prizes y = (x∗/(n− 1), . . . , x∗/(n− 1), 0) whenever u is strictly concave. Bycontradiction, let Cy be a contest that implements (e∗, ..., e∗) with

∑nk=1 yk = x∗ and yn = 0

but y1 6= yn−1. Proceeding as before, summing the inequalities Πi((e∗, . . . , e∗), Cy) ≥ 0

over all agents yields∑n−1

k=1 u(yk)−nc(e∗) ≥ 0. Strict concavity of u, however, implies that∑n−1k=1 u(yk)− nc(e∗) < (n− 1)u(x∗/(n− 1))− nc(e∗) = 0, a contradiction.If-statement. We showed above that the upper bound on the principal’s payoff is given

by ne∗ − x∗. Thus, any contest which implements (e∗, ..., e∗) with the prize sum x∗ attainsthe upper bound. �

A.3 Proof of Corollary 1

Each optimal contest induces individual efforts of e∗ and pays a sum of x∗, as characterizedin Theorem 2. Now consider the principal’s first-best problem. If the agents’ efforts weredirectly observable and verifiable, then the principal could ask for individual efforts of e andwould have to compensate the agents with a transfer sum x such that u(x/n) − c(e) = 0.

46

Put differently, for a given transfer sum x the maximal achievable individual effort is

ex = c−1(u(xn

)),

and the first-best problem is to maximize nex − x by choice of x ∈ [0, T ]. With the samearguments as in the proof of Theorem 2, this yields xFB = min{x, T}, where x is given by

u′(x

n

)= c′

(c−1

(u

(x

n

))).

The resulting optimal effort level is

eFB = c−1

(u

(xFB

n

)).

Now suppose that the agents are risk-neutral, i.e., the function u is linear. The conditionscharacterizing (e∗, x∗) in Theorem 2 then coincide with those characterizing (eFB, xFB)

above, which implies (e∗, x∗) = (eFB, xFB). Then suppose that the agents are risk-averse,i.e., the function u is strictly concave. If x∗ 6= xFB there is nothing to prove. Hence assumex∗ = xFB. Inspection of the conditions that define e∗ and eFB then immediately revealsthat e∗ < eFB. �

A.4 Proof of Theorem 3

The proof proceeds in two steps. Step 1 shows that (e∗, . . . , e∗) is an equilibrium of thecontest C∗y described in the theorem. Step 2 shows that no other equilibria exist. Thestructure of the arguments in Step 2 is reminiscent of equilibrium characterization proofsin all-pay auctions with or without caps (e.g. Baye, Kovenock, and De Vries, 1996, or Cheand Gale, 1998). Throughout the proof, we adopt the interpretation of a non-physical cap,so efforts ei > e∗ are possible but are not differentiated by the reviewer. The result thenalso follows for the case where e∗ is a physical bound on efforts.

Step 1. Consider deviations e′i of agent i from (e∗, . . . , e∗). If e′i > e∗, we obtain

Πi((e∗, . . . , e∗), C∗y ) =

n− 1

nu

(x∗

n− 1

)− c(e∗)

>n− 1

nu

(x∗

n− 1

)− c(e′i)

= Πi((e∗, . . . , e′i, . . . , e

∗), C∗y ).

47

If e′i < e∗, we obtain

Πi((e∗, . . . , e∗), C∗y ) =

n− 1

nu

(x∗

n− 1

)− c(e∗)

= 0 ≥ −c(e′i) = Πi((e∗, . . . , e′i, . . . , e

∗), C∗y ).

Thus, the contest C∗y implements the effort profile (e∗, ..., e∗).Step 2. By contradiction, suppose C∗y also implements some other profile σ 6= (e∗, . . . , e∗).

Denote the support of σi by Li, so ei ∈ Li if and only if every open neighbourhood N of eisatisfies σi(N) > 0. We first show that it must be that Li ⊆ [0, e∗] for all i ∈ I. Suppose not,so there exists an agent i and an effort level ei > e∗ such that σi((ei−ε, ei+ε)) > 0 ∀ε > 0.Fix ε > 0 such that ei − ε > e∗. Note that the expected payoff of agent i playing e′i ≥ e∗

with probability one, while the other agents play σ−i, is

Πi(e′i, σ−i) =

[1−

∏j 6=i

σj([e∗,∞)) +

∏j 6=i

σj([e∗,∞))

n− 1

n

]u

(x∗

n− 1

)− c(e′i).

where we omit the dependence on C∗y to simplify notation. Since c is strictly increasing,we have Πi(e

∗, σ−i) > Πi(e′i, σ−i) for all e′i > e∗. Hence Πi(e

∗, σ−i) > Πi(ei, σ−i) for allei ∈ (ei− ε, ei + ε). Since σi((ei− ε, ei + ε)) > 0, agent i could strictly increase his expectedpayoff by shifting the mass from this interval to e∗. Thus, σ is not an equilibrium. Fromnow on, we only consider the cases where Li ⊆ [0, e∗] ∀i ∈ I. Let ei = minLi. Since theproposed profile σ is different from (e∗, ..., e∗), it must be that e = mini∈I ei < e∗.

First, suppose that e > 0. Furthermore suppose that σj({e}) > 0 for exactly one agentj ∈ I, or that σi({e}) = 0 for all i ∈ I. In the latter case let j be such that ej = e. Thenthere exists some ε > 0 such that

Πj(e+ ε, σ−j) ≤

[1−

∏i 6=j

σi((e+ ε,∞))

]u

(x∗

n− 1

)− c(e+ ε) < 0

for all ε < ε. Intuitively, the probability that agent j wins a positive prize approacheszero as ε approaches zero (by right continuity of σi((e + ε,∞)) in ε and σi((e,∞)) = 1),while the cost of effort at e is strictly positive. Hence agent j could strictly increase hisexpected payoff by shifting the mass σj([e, e + ε)) > 0 from [e, e + ε) to 0. Next supposethat σi({e}) > 0 for at least two agents i = j, k. Then there exists a small ε > 0 such that

Πj(e, σ−j) ≤

[1−

(1− 1

2σk({e})

) ∏i 6=j,k

σi((e,∞))

]u

(x∗

n− 1

)− c(e)

<

[1−

∏i 6=j

σi((e,∞))

]u

(x∗

n− 1

)− c(e+ ε)

48

≤ Πj(e+ ε, σ−j).

The intuition is that a small upward deviation from e increases the probability of winningdiscretely, while marginally increasing the effort costs. Hence agent j could strictly increasehis expected payoff by shifting the mass σj({e}) > 0 from e to e + ε. We conclude thatthere does not exist an equilibrium σ 6= (e∗, ..., e∗) with e > 0.

Second, suppose that e = 0. Consider first the case where σi({0}) = 0 for all i ∈ I,that is, no agent places an atom on 0. If there is an agent j such that ek > 0 for all k 6= j,then there exists some ε > 0 such that Πj(ε, σ−j) = −c(ε) for all ε < ε. Agent j couldthen strictly increase his expected payoff by shifting the mass σj((0, ε)) > 0 from (0, ε) to0. Thus, there have to be at least two agents j and k with ej = ek = 0. But in this case,observe that

Πj(ε, σ−j) ≤

[(1−

∏i 6=j

σi((ε,∞))

)u

(x∗

n− 1

)− c(ε)

]

and

limε→0

[(1−

∏i 6=j

σi((ε,∞))

)u

(x∗

n− 1

)− c(ε)

]= 0.

Thus for every Π > 0 there exists ε > 0 such that Πj(ε, σ−j) < Π for all ε < ε. Intuitively,both the probability of winning and the costs approach zero as ε→ 0. However, it must bethat Πj(e

∗, σ−j) > 0 since Πj(e∗, . . . , e∗) = 0 and the probability that j wins a positive prize

is strictly greater if the other agents play σ−j, because at least agent k exerts efforts lowerthan e∗ with strictly positive probability. Hence agent j could strictly increase his expectedpayoff by shifting the mass σj((0, ε)) > 0 from (0, ε) to e∗, for some sufficiently small ε > 0.The only remaining case is σj({0}) > 0 for at least one agent j ∈ I. Observe that there canonly be one such agent, since otherwise a small upward deviation from 0 would lead to adiscrete increase in the probability of winning a positive prize, analogous to the argumentabove. Then it must be that Πj(σj, σ−j) = 0 since Πj(0, σ−j) = 0. This can only be themaximum payoff of agent j if all other agents exert deterministic efforts equal to e∗, sinceotherwise Πj(e

∗, σ−j) > 0. In this case, agent j is indifferent between playing 0 or e∗, andall other effort levels yield strictly lower payoffs. This implies σj({0}) +σj({e∗}) = 1. Nowconsider an agent k 6= j. Observe that a deviation by agent k to some ε with 0 < ε < e∗

leads to payoffs

Πk(ε, σ−k) = σj({0})u(

x∗

n− 1

)− c(ε).

49

Thus a sufficiently small ε > 0 will be a profitable deviation whenever

σj({0})u(

x∗

n− 1

)> σj({0})u

(x∗

n− 1

)+ (1− σj({0}))

n− 1

nu

(x∗

n− 1

)− c(e∗).

This can be reformulated to

0 > −σj({0})n− 1

nu

(x∗

n− 1

),

which always holds because σj({0}) > 0. �

B Proofs of Additional Results

B.1 Proof of Proposition 1

Fix an arbitrary observational structure (S, η) and a contract Ψ. It is an easy exerciseto check that the equivalents of Lemmas 1 - 3 hold in the general setting. It is thereforewithout loss of generality to assume that Ψ = (ψs)s∈S does not screen the reviewer’s type,and that there exist x, x ∈ R+ such that

Eψs[

n∑i=1

ti

]= x, Eψs

[n∑i=1

u(ti)

]= x

for all s ∈ S.Now suppose that a contest Ky with prize profile y = (x∗/(n − 1), . . . , x∗/(n − 1), 0)

implements (e∗, ..., e∗). By contradiction, assume that Ky is not optimal, i.e., there existsa contract Ψ = (ψs)s∈S that implements some strategy profile σ and

ΠP (σ,Ψ) = Eσ

[n∑i=1

ei

]− Eσ

[Eηe

[Eψs

[n∑i=1

ti

]]]> ΠP ((e∗, . . . , e∗), Ky) = ne∗ − x∗.

Construct a (non-screening) contract Φ = (µe)e∈E for the setting with perfect observationby defining

µe(A) = Eηe [ψs (A)]

for each measurable subset A ⊆ T and all e ∈ E. We then obtain

Eµe[

n∑i=1

ti

]= Eηe

[Eψs

[n∑i=1

ti

]]= Eηe [x] = x,

50

Eµe[

n∑i=1

u(ti)

]= Eηe

[Eψs

[n∑i=1

u(ti)

]]= Eηe [x] = x,

for some x, x ∈ R+ and all e ∈ E. Hence Φ is credible. It also holds that

Πi(σ′,Φ) = Eσ′ [Eµe [u(ti)]]− Eσ′i [c(ei)]

= Eσ′ [Eηe [Eψs [u(ti)]]]− Eσ′i [c(ei)]

= Πi(σ′,Ψ)

for all profiles σ′ and all i ∈ I. Hence Φ implements σ with perfect observation because Ψ

implements σ with imperfect observation. We finally obtain

ΠP (σ,Φ) = Eσ

[n∑i=1

ei

]− Eσ

[Eµe

[n∑i=1

ti

]]

= Eσ

[n∑i=1

ei

]− Eσ

[Eηe

[Eψs

[n∑i=1

ti

]]]= ΠP (σ,Ψ) > ne∗ − x∗,

in contradiction to Theorem 2. �

B.2 Examples of Imperfect Effort Observation

B.2.1 First Example

Consider the moral-hazard example with log-normal noise. Suppose that the conditionσ2

1 + σ22 − 2σ12 ≤ 2/(πβ2) is satisfied. Let Ky be the contest with prize profile (x∗, 0) in

which x∗ is given to agent 1 if and only if rs1/s2 ≥ 1, where r ∼ lnN [νr, σ2r ] is a log-normal

random variable with parameters

νr = ν2 − ν1 and σ2r =

2

πβ2− (σ2

1 + σ22 − 2σ12).

This allows for σ2r = 0, by which we mean that r is degenerate and takes the value eνr with

probability one. We now proceed in two steps. Step 1 derives an expression for agent i’sexpected payoff as a function of the effort profile e. Step 2 shows that ei = e∗ is a bestresponse when agent j 6= i chooses ej = e∗.

Step 1. Given an effort profile e, the probability that agent 1 wins the prize is

p(e) = Pr[rs1

s2

≥ 1

]= Pr

[rr1e1

r2e2

≥ 1

]= Pr

[r2

rr1

≤ e1

e2

].

51

Since the variables r1, r2 and r are log-normally distributed, it follows that the compoundvariable r2/(rr1) is also log-normal, with location parameter ν = ν2− ν1− νr = 0 and scaleparameter σ2 = σ2

1 + σ22 − σ12 + σ2

r = 2/(πβ2). The cdf of the log-normal distribution isgiven by F (x) = Φ ((log x− ν)/σ), where Φ is the cdf of the standard normal distribution.Thus we can write

p(e) = Φ

(log(e1/e2)β

√π

2

).

For the probability that agent 2 wins the prize we obtain

1− p(e) = 1− Φ

(log(e1/e2)β

√π

2

)= Φ

(− log(e1/e2)β

√π

2

)= Φ

(log(e2/e1)β

√π

2

).

Hence the expected payoff of agent i = 1, 2 is

Πi(e) = Φ

(log(ei/ej)β

√π

2

)u∗ − c(ei)

= Φ

(log(ei/ej)β

√π

2

)2γe∗β − γeβi .

Step 2. Suppose ej = e∗ and consider the choice of agent i 6= j. We immediately obtainΠi(e

∗, e∗) = 0. We will now show that Πi(ei, e∗) ≤ 0 always holds, i.e.,

Φ

(log(ei/e

∗)β

√π

2

)≤ 1

2

( eie∗

)βfor all ei ∈ R+. After the change of variables x = log(ei/e

∗)β√π/2 this becomes the

requirement that

Φ(x) ≤ 1

2ex√

2/π (3)

for all x ∈ R. Inequality (3) is satisfied for x = 0, where LHS and RHS both take a value of1/2. Furthermore, the LHS function and the RHS function are tangent at x = 0, becausetheir derivatives are both equal to 1/

√2π at this point. It then follows immediately that

inequality (3) is also satisfied for all x > 0, because the LHS is strictly concave in x in thisrange, while the RHS is strictly convex. We now consider the remaining case where x < 0.

52

We use the fact that Φ(x) = erfc(−x/√

2)/2, where

erfc(y) =2√π

∫ ∞y

e−t2

dt

is the complementary error function (see e.g. Chang, Cosman, and Milstein, 2011). Afterthe change of variables y = −x/

√2 we thus need to verify

erfc(y) ≤ e−2y/√π (4)

for all y > 0. Inequality (4) is satisfied for y = 0, where LHS and RHS both take a valueof 1. Now observe that the derivative of the LHS with respect to y is given by −2e−y

2/√π,

while the derivative of the RHS is −2e−2y/√π/√π. The condition that the former is weakly

smaller than the latter can be rearranged to y ≤ 2/√π, which implies that (4) is satisfied for

0 < y ≤ 2/√π. For larger values of y, we can use a Chernoff bound for the complementary

error function. Theorem 1 in Chang et al. (2011) implies that

erfc(y) ≤ e−y2

for all y ≥ 0. The inequality e−y2 ≤ e−2y/√π can be rearranged to y ≥ 2/

√π. This implies

that (4) is satisfied also for y > 2/√π.

B.2.2 Second Example

Consider the example where only the effort difference s = e1 − e2 can be observed. LetKy be the contest with prize profile (x∗, 0) in which x∗ is given to agent 1 if and only ifs+ r ≥ 0, where r ∼ U [−c(e∗)/c′(e∗), c(e∗)/c′(e∗)] is a uniform random variable.

Observe that c(e∗)/c′(e∗) < e∗ holds due to strict convexity of c and c(0) = 0. We cantherefore write the probability that agent 1 wins the prize, holding the effort e2 = e∗ fixed,as a piecewise function

p(e1) =

1 if e1 > e∗ + c(e∗)

c′(e∗),

12

+ 12c′(e∗)c(e∗)

(e1 − e∗) if e∗ − c(e∗)c′(e∗)

≤ e1 ≤ e∗ + c(e∗)c′(e∗)

,

0 if e1 < e∗ − c(e∗)c′(e∗)

.

Then, the expected payoff of agent 1 is given by

Π1(e1) = p(e1)u∗ − c(e1) = p(e1)2c(e∗)− c(e1).

It follows that Π1(e∗) = 0. We now consider the three types of deviations from e∗.Case 1: e1 < e∗ − c(e∗)/c′(e∗). It follows immediately that Π1(e1) ≤ 0 in this range,

53

which implies that these deviations are not profitable.Case 2: e∗− c(e∗)/c′(e∗) ≤ e1 ≤ e∗+ c(e∗)/c′(e∗). Observe that Π′1(e1) = c′(e∗)− c′(e1)

in this range. Hence the first-order condition yields the unique solution e1 = e∗. SinceΠ′′1(e1) = −c′′(e1) < 0, this is indeed the maximum over this range.

Case 3: e1 > e∗ + c(e∗)/c′(e∗). We have Π1(e1) < Π1(e∗ + c(e∗)/c′(e∗)) for this range.Hence, by the arguments for the previous case, these deviations are not profitable either.

We conclude that e1 = e∗ is a best response to e2 = e∗. The argument for agent 2 issymmetric, which implies that the contest implements (e∗, e∗).

B.3 Proof of Proposition 2

We first derive some results which hold under cost heterogeneity for any number of agents.It is easy to see that Lemmas 1 to 4 continue to hold. Hence it is still without loss ofgenerality to consider a contract Φ = (µe)e∈E that does not screen θ, satisfies

Eµe[

n∑i=1

ti

]= x and Eµe

[n∑i=1

u(ti)

]= x

for all e ∈ E, and implements a possibly asymmetric pure effort profile e = (e1, . . . , en).For any such contract, we obtain the following intermediate result.

Lemma 7 If a contract Φ implements a pure-strategy effort profile e = (e1, . . . , en), then

1

n− 1

n∑i=1

ci(ei) ≤ u

(x

n− 1

).

Proof. Since Φ implements e, for each i ∈ I it must hold that

ci(ei) ≤ Eµe [u(ti)]− Eµ(0,e−i) [u(ti)] .

Summing over all i ∈ I, we obtain

n∑i=1

ci(ei) ≤n∑i=1

Eµe [u(ti)]−n∑i=1

Eµ(0,e−i) [u(ti)]

= x−

(x−

n∑i=2

Eµ(0,e−1) [u(ti)]

)−

n∑i=2

Eµ(0,e−i) [u(ti)]

≤n∑i=2

Eµ(0,e−1) [u(ti)]

≤n∑i=2

u(Eµ(0,e−1) [ti]

)

54

≤n∑i=2

u

(x

n− 1

)= (n− 1)u

(x

n− 1

),

where the third inequality follows from concavity of u, and the fourth inequality followsfrom concavity of u together with the fact that

∑ni=2 Eµ(0,e−1) [ti] ≤ x. �

For the special case of n = 2, we can now replace the previous Lemmas 5 and 6 by thefollowing result.

Lemma 8 Suppose n = 2. For every contract Φ that implements a pure-strategy effortprofile e, there exists a contest Cy that also implements e and yields the same expectedpayoff to the principal.

Proof. Suppose Φ implements e. By Lemma 7 it holds that c1(e1) + c2(e2) ≤ u(x), so inparticular ci(ei) ≤ u(x) for both i = 1, 2. We now construct a contest Cy with prize profiley = (x, 0) that also implements e. The case where x = 0 and thus e1 = e2 = 0 is trivial, sowe focus on the case where x > 0.

The allocation rule of Cy is as follows. If e = e, the zero prize is given to agent iwith probability pi ≥ 0, while the other agent obtains the positive prize. Below we willdetermine the values pi such that p1 + p2 = 1. If e = (ei, e−i) with ei 6= ei, the deviatingagent i obtains the zero prize for sure and the non-deviating agent obtains the positiveprize. For all other effort profiles e, the allocation of the prizes can be chosen arbitrarily.Since Cy is a contest, it is credible.

For each agent i = 1, 2, first define pi implicitly by

(1− pi)u (x) = ci(ei).

Since the LHS of this equation describes the expected payoff of agent i who expects toobtain the zero prize with probability pi, it follows that the contest Cy indeed implementse if pi ≤ pi holds for both i = 1, 2. The fact that ci(ei) ≤ u(x) guarantees pi ≥ 0 for bothi = 1, 2. Lemma 7 also implies that

2∑i=1

ci(ei) =2∑i=1

(1− pi)u(x) = (2− p1 − p2)u(x) ≤ u(x),

which guarantees that p1 + p2 ≥ 1. It is therefore possible to find equilibrium punishmentprobabilities pi such that 0 ≤ pi ≤ pi and p1 + p2 = 1.

Finally, from the above arguments we also obtain that the principal’s expected payoffis e1 + e2 − x with both (e,Φ) and (e, Cy). �

55

Since the principal can achieve the same payoff with a contest as with any other con-tract, it only remains to be shown that a solution to the principal’s problem exists in theheterogeneous cost case. This will be established in the proof of Proposition 4 below. �

B.4 Proof of Proposition 3

The proof proceeds in two steps. First, we state some properties that must be satisfied bythe optimal contract for any given profile of effort cost functions. Second, we prove themain statement about the optimality of contests for large m.

Step 1. Fix any profile of cost functions (c1, . . . , cn). As argued in the proof of Proposi-tion 2, we can restrict attention to non-screening contracts with expected sums of transfers(x) and expected sums of utilities (x) that are fixed across e ∈ E, and pure-strategy effortprofiles. The following result provides a lower bound on the principal’s maximal profitswithin that class. Define the bound

Π = maxx∈[0,T ]

[c−1

1 (u(x))− x],

which satisfies Π > 0 due to our assumptions on c1 and u.

Lemma 9 There exists a contest Cy that implements some effort profile e ∈ E such thatΠP (e, Cy) = Π.

Proof. Let x∗ = arg maxx∈[0,T ]

[c−1

1 (u(x))− x]and e∗1 = c−1

1 (u(x∗)). Consider a contestCy with prize profile y = (x∗, 0, . . . , 0). If the effort profile e is such that e1 = e∗1, thenagent 1 receives the prize x∗ while all other agents receive a zero prize. For any other effortprofile, agent 2 receives x∗ and all other agents receive a zero prize. It follows immediatelythat Cy implements (e∗1, 0, . . . , 0) and yields the payoff e∗1 − x∗ = Π to the principal. �

The next result is a direct generalization of Lemma 5 to the case of heterogeneous costfunctions. The proof proceeds exactly like the proof of Lemma 5 and is therefore omitted.

Lemma 10 For every contract Φ that implements a pure-strategy profile e = (e1, ..., en)

such that

1

n

n∑i=1

ci(ei) > ck

(1

n

n∑i=1

ei

)∀k ∈ I,

there exists a contract Φ that implements the pure-strategy profile e = (e1, ..., en), wheree1 = . . . = en = 1

n

∑ni=1 ei, and yields the same expected payoff to the principal.

To summarize, in the search for an optimal contract we can restrict attention to non-screening contracts Φ with an expected sum of transfers x and an expected sum of utilities

56

x that are fixed across all e ∈ E, and which implement a pure-strategy effort profile e suchthat ΠP (e,Φ) ≥ Π and

1

n

n∑i=1

ci(ei) ≤ maxk∈I

ck

(1

n

n∑i=1

ei

). (5)

Step 2. Consider any sequence (cm1 , . . . , cmn )m∈N where, for each agent i ∈ I, the

cost functions cmi converge uniformly to the common cost function c as m → ∞. Let(em,Φm)m∈N be an arbitrary sequence such that contract Φm implements effort profile em =

(em1 , . . . , emn ) when the cost functions are (cm1 , . . . , c

mn ). We will write em = (1/n)

∑ni=1 e

mi

for the average effort at step m in the sequence. Without loss of generality, assume thatthe conditions summarized at the end of Step 1 are satisfied for each m ∈ N.

Lemma 11 The sequence

κm = maxk∈I

cmk (emk )− 1

n

n∑i=1

cmi (emi )

converges to zero as m→∞.

Proof. For every m ∈ N, let

δm = maxk∈I

cmk (em)− 1

n

n∑i=1

cmi (emi ) and ψm = maxi∈I

cmi (emi )−maxk∈I

cmk (em),

and hence κm = δm + ψm. We will show that limm→∞ δm = limm→∞ ψ

m = 0, whichimmediately implies that limm→∞ κ

m = 0.For the sequence δm, note that

δm =

[maxk∈I

cmk (em)− c(em)

]+

[1

n

n∑i=1

c(emi )− 1

n

n∑i=1

cmi (emi )

]+

[c(em)− 1

n

n∑i=1

c(emi )

].

By uniform convergence of cmi to c, ∀i ∈ I, we have

limm→∞

(cmi (em)− c(em)) = 0 and limm→∞

(cmi (emi )− c(emi )) = 0 ∀i ∈ I, (6)

and thus

limm→∞

maxk∈I

(cmk (em)− c(em)) = 0 and limm→∞

(1

n

n∑i=1

cmi (emi )− 1

n

n∑i=1

c(emi )

)= 0.

In addition, by convexity of c we have c(em) − 1n

∑ni=1 c(e

mi ) ≤ 0 for all m ∈ N, and by

57

condition (5) we have δm ≥ 0 for all m ∈ N. Hence, we must also have

limm→∞

(c(em)− 1

n

n∑i=1

c(emi )

)= 0, (7)

as otherwise for some large m we would have δm < 0, a contradiction. This concludes thatlimm→∞ δ

m = 0.For the sequence ψm, we have

ψm = maxk∈I

(c(em)− cmk (em)) + maxi∈I

[cmi (emi )− c(emi ) + c(emi )− c(em)] .

Hence, by (6), a sufficient condition for limm→∞ ψm = 0 is

limm→∞

(c(emi )− c(em)) = 0 ∀i ∈ I. (8)

To establish (8), we first claim that there exists e > 0 such that emi ∈ [0, e] for all i ∈ I andall m ∈ N. The fact that Φm implements em implies cmi (emi ) ≤ u(T ) for all i ∈ I. Now fixany u > u(T ). By uniform convergence of each cmi to c it follows that there exists m′ ∈ Nsuch that for all m ≥ m′,

|cmi (emi )− c(emi )| ≤ u− u(T ) ∀i ∈ I,

which then implies c(emi ) ≤ u and therefore emi ≤ c−1(u). Now just define e as the maximumamong c−1(u) and the finite number of values emi for all i ∈ I and m < m′. We next claimthat limm→∞(emi − em) = 0 holds for all i ∈ I. By contradiction, assume there existsi ∈ I and ε > 0 such that for all m′ ∈ N there exists m ≥ m′ so that |emi − em| ≥ ε.Define Ei = {(e1, . . . , en) ∈ [0, e]n | |ei − 1

n

∑nj=1 ej| ≥ ε}. The set Ei is compact and the

function χ(e) = 1n

∑nj=1 c(ej) − c

(1n

∑nj=1 ej

)is continuous on Ei, with χ(e) > 0 due to

strict convexity of c and ε > 0. Hence ε = mine∈Ei χ(e) exists and satisfies ε > 0. Wehave thus shown that there exists ε > 0 such that for all m′ ∈ N there exists m ≥ m′

so that χ(em) = −(c(em) − 1n

∑ni=1 c(e

mi )) ≥ ε, contradicting (7). Finally, (8) now follows

immediately because emi ∈ [0, e] and em ∈ [0, e] and c is continuous on [0, e]. �

Next we show that the sum of effort costs is bounded away from zero for large m.

Lemma 12 There exist m′ ∈ N and c > 0 such that∑n

i=1 cmi (emi ) ≥ c for all m ≥ m′.

Proof. Let Πm = maxx∈[0,T ] Πm1 (x) with Πm

1 (x) = (cm1 )−1(u(x)) − x be the lower profitbound for the cost functions (cm1 , . . . , c

mn ) as defined in Step 1 of the proof. Hence we know

that ΠP (em,Φm) ≥ Πm holds for all m ∈ N. Similarly, let Π∞ = maxx∈[0,T ] Π1(x) withΠ1(x) = c−1(u(x))− x be the bound when the cost functions are (c, . . . , c). We first claim

58

that limm→∞Πm = Π∞. The claim follows immediately once we show that Πm1 converges

uniformly to Π1 on [0, T ]. Using Theorem 2 in Barvinek, Daler, and Francu (1991), it canbe shown that (cm1 )−1 converges uniformly to c−1 on [0, u(T )].39 Thus for every ε > 0 thereexists m′′ ∈ N such that for all m ≥ m′′,

|Πm1 (x)− Π1(x)| = |(cm1 )−1(u(x))− c−1(u(x))| < ε

for all x ∈ [0, T ], which establishes uniform convergence.Now fix any ε such that 0 < ε < Π∞ and define Π = Π∞ − ε > 0. Hence there exists

m′′′ ∈ N such that for all m ≥ m′′′,

n∑i=1

emi ≥ ΠP (em,Φm) ≥ Πm ≥ Π > 0.

Define

cm = mine∈E

n∑i=1

cmi (ei) s.t.n∑i=1

ei = Π.

We then obtain that∑n

i=1 cmi (emi ) ≥ cm for all m ≥ m′′′. Similarly, define

c∞ = mine∈E

n∑i=1

c(ei) s.t.n∑i=1

ei = Π,

noting that c∞ > 0. It again follows from uniform convergence of cmi to c for each i ∈ Ithat limm→∞ c

m = c∞. Fix any ε′ such that 0 < ε′ < c∞ and define c = c∞ − ε′ > 0. Itfollows that there exists m′ ∈ N such that for all m ≥ m′,

n∑i=1

cmi (emi ) ≥ cm ≥ c,

which completes the proof. �

We can now combine Lemmas 11 and 12 to obtain the following result.

Lemma 13 There exists m ∈ N such that for all m ≥ m,

maxk∈I

cmk (emk ) ≤ 1

n− 1

n∑i=1

cmi (emi ).

Proof. By Lemma 12, there exist m′ ∈ N and c > 0 such that∑n

i=1 cmi (emi ) ≥ c for all

m ≥ m′. In addition, from the limiting statement about κm in Lemma 11 we can conclude39 The theorem is directly applicable and implies our claim after we extend the functions c and cm1 to

R by defining cm1 (e) = c(e) = e for all e < 0.

59

that there exists m′′ ∈ N such that for all m ≥ m′′,

maxk∈I

cmk (emk )− 1

n

n∑i=1

cmi (emi ) ≤ c

n(n− 1).

Thus for all m ≥ m = max{m′,m′′} we obtain

maxk∈I

cmk (emk )− 1

n− 1

n∑i=1

cmi (emi ) = maxk∈I

cmk (emk )− 1

n

n∑i=1

cmi (emi )− 1

n(n− 1)

n∑i=1

cmi (emi )

≤ c

n(n− 1)− 1

n(n− 1)

n∑i=1

cmi (emi )

≤ 0. �

Now consider the sequence for any m ≥ m as given in Lemma 13. Combined withLemma 7 we can conclude that

maxk∈I

cmk (emk ) ≤ u

(xm

n− 1

)(9)

holds, where xm > 0 is the expected sum of transfers in contract Φm. We now claim thatthere exists a contest Cm

y which also implements em and yields the same expected payoffto the principal. Let the prize profile be given by

y =

(xm

n− 1, . . . ,

xm

n− 1, 0

).

The allocation rule of Cmy is as follows. If e = em, the zero prize is given to agent i with

probability pmi ≥ 0, while all other agents obtain one of the identical positive prizes. Belowwe will determine the values pmi such that

∑ni=1 p

mi = 1. If e = (ei, e

m−i) with ei 6= emi for

some i ∈ I, the deviating agent i obtains the zero prize for sure and all other agents obtainone of the identical positive prizes. For all other effort profiles e, the allocation of the prizescan be chosen arbitrarily. Since Cm

y is a contest, it is credible.For each agent i ∈ I, first define pmi implicitly by

(1− pmi )u

(xm

n− 1

)= cmi (emi ).

Since the LHS of this equation describes the expected payoff of agent i who expects to obtainthe zero prize with probability pmi , it follows that the contest Cm

y indeed implements em ifpmi ≤ pmi holds for all i ∈ I. The fact that cmi (emi ) ≤ u(xm/(n− 1)) for all i ∈ I due to that(9) guarantees pmi ≥ 0. Lemma 7 also implies that

n∑i=1

cmi (emi ) =n∑i=1

(1− pmi )u

(xm

n− 1

)=

(n−

n∑i=1

pmi

)u

(xm

n− 1

)≤ (n− 1)u

(xm

n− 1

),

60

which guarantees that∑n

i=1 pmi ≥ 1. It is therefore possible to find equilibrium punishment

probabilities pmi such that 0 ≤ pmi ≤ pmi ∀i ∈ I and∑n

i=1 pmi = 1. The principal’s expected

payoff is then∑n

i=1 emi − xm with both (em,Φm) and (em, Cm

y ).In sum, for any m ≥ m, the principal can achieve a weakly higher payoff with a contest

than with any other contract. Hence it only remains to be shown that a solution to theprincipal’s problem exists in the heterogeneous cost case. This will be established in theproof of Proposition 4 below. �

B.5 Proof of Proposition 4

First, we characterize the set of all pure-strategy effort profiles which can be implemented.Define

E1 =

{e ∈ E |

n∑i=1

ci(ei) ≤ (n− 1)u

(T

n− 1

)},

E2 =

{e ∈ E |

n∑i=1

u−1(ci(ei)) ≤ T

}.

The following result provides a necessary condition for implementability.

Lemma 14 A pure-strategy effort profile e ∈ E can be implemented by some contract onlyif e ∈ E1 ∩ E2.

Proof. Suppose e ∈ E is implemented by a contract Φ. By contradiction, assume firstthat e /∈ E1. Then

∑ni=1 ci(ei) > (n − 1)u(T/(n − 1)). By Lemma 7, it also holds

that∑n

i=1 ci(ei) ≤ (n − 1)u (x/(n− 1)), where x is the expected sum of transfers in Φ.Combining the two inequalities, we obtain x > T . Hence Φ allocates transfer profileswhich are not feasible, a contradiction.

Suppose now that e /∈ E2. From (IC-A) it is clear that ci(ei) ≤ Eµe [u(ti)] ≤ u(Eµe [ti])

must hold, and hence u−1(ci(ei)) ≤ Eµe [ti], for all i ∈ I. Since e /∈ E2, we then obtainx =

∑ni=1 Eµe [ti] > T , again a contradiction. �

We now restrict attention to effort profiles e ∈ E1 ∩ E2, assuming without loss ofgenerality that ci(ei) ≥ cj(ej) if i < j. For any such profile, define x′ implicitly by

u

(x′

n− 1

)=

1

n− 1

n∑i=1

ci(ei).

61

The fact that e ∈ E1 implies that x′ is well-defined and satisfies x′ ≤ T . Furthermore, foreach i ∈ I let xi = u−1(ci(ei)) and

x =n∑i=1

xi,

which is well-defined and satisfies x ≤ T because e ∈ E2.

Lemma 15 If e ∈ E1 ∩ E2, then e can be implemented by a generalized contest with anexpected sum of transfers max{x′, x}.

Proof. Fix any e ∈ E1 ∩ E2. We will construct a generalized contest that implements ewith an expected sum of transfers max{x′, x} ≤ T . We examine two cases separately.

Case 1: x′ ≥ x. Define yd = (x′/(n − 1), . . . , x′/(n − 1), 0). This vector of prizes willbe used after any unilateral deviation from e, so that the deviator gets the zero prize whileall other agents get x′/(n− 1). Next, we construct the vector of prizes y that will be usedto reward the agents in equilibrium. For every ε ∈ [0, xn], let

yn(ε) = xn − ε and yi(ε) = xi +x′ − x+ ε

n− 1∀i < n.

By construction∑n

i=1 yi(ε) = x′. In addition, we have

n∑i=1

u(yi(0)) ≥n∑i=1

ci(ei) = (n− 1)u

(x′

n− 1

)≥

n∑i=1

u(yi(xn)), (10)

where the first inequality follows from yi(0) ≥ xi for all i ∈ I, and the second inequalityfollows from yn(xn) = 0,

∑ni=1 yi(xn) = x′, and the fact that u is concave. Since u is

continuous, (10) implies that there exists some ε∗ ∈ [0, xn] such that

n∑i=1

u(yi(ε∗)) =

n∑i=1

ci(ei) = (n− 1)u

(x′

n− 1

). (11)

Let y = y(ε∗) = (y1(ε∗), ..., yn(ε∗)). It follows by construction that any contract whichallocates only permutations of the vectors y or yd is credible.

Since the transfer to a unilateral deviator from e is zero, the only remaining step is toshow that there is a way to allocate the equilibrium prizes y such that the expected utilityof each agent i is at least ci(ei).

62

Consider the following n transfer vectors, each of which is a permutation of y. For alli ∈ I, ti ∈ T is given by

tij =

yn(ε∗) if j = i,

yi(ε∗) if j = n,

yj(ε∗) if j 6= i, n.

In words, in transfer vector tn each agent j receives the transfer yj(ε∗). For any i 6= n, thetransfer vector ti is the same as tn except that the transfers of agents i and n are swapped.Next, we will construct the probabilities with which the transfer vectors ti are allocated inequilibrium.

If ε∗ = 0, then we have u(yi(ε∗)) ≥ ci(ei) for all i ∈ I. In this case, we let the transfer

vector tn be allocated with probability one in equilibrium, so that the required expectedutility is achieved for each i ∈ I.

If ε∗ > 0, then we have u(yi(ε∗)) > ci(ei) ≥ cn(en) > u(yn(ε∗)) for all i < n. Hence, for

all i < n there exists a unique pi ∈ (0, 1) such that

piu(yn(ε∗)) + (1− pi)u(yi(ε∗)) = ci(ei). (12)

In addition, we define pn = 1−∑n−1

i=1 pi. To show that (p1, ..., pn) is a well-defined proba-

bility vector, we only need to establish pn ≥ 0. This property indeed holds because

n−1∑i=1

pi =n−1∑i=1

u(yi(ε∗))− ci(ei)

u(yi(ε∗))− u(yn(ε∗))

≤n−1∑i=1

u(yi(ε∗))− ci(ei)

u(yn−1(ε∗))− u(yn(ε∗))

=

∑ni=1 u(yi(ε

∗))− u(yn(ε∗))−∑n−1

i=1 ci(ei)

u(yn−1(ε∗))− u(yn(ε∗))

=

∑ni=1 ci(ei)− u(yn(ε∗))−

∑n−1i=1 ci(ei)

u(yn−1(ε∗))− u(yn(ε∗))

=cn(en)− u(yn(ε∗))

u(yn−1(ε∗))− u(yn(ε∗))

< 1.

Now consider the allocation rule where each vector ti is allocated with probability pi.With this rule, each agent i < n receives the prize yn(ε∗) with probability pi and the prizeyi(ε

∗) with the opposite probability 1 − pi. By (12), this yields an expected utility equalexactly to ci(ei). As for agent n, he receives each prize yi(ε∗) with probability pi. Hence,

63

his expected utility is given by

n∑i=1

piu(yi(ε∗)) =

n−1∑i=1

piu(yi(ε∗)) +

(1−

n−1∑i=1

pi

)u(yn(ε∗))

=n−1∑i=1

pi [u(yi(ε∗))− u(yn(ε∗))] + u(yn(ε∗))

=n−1∑i=1

[u(yi(ε∗))− ci(ei)] + u(yn(ε∗))

=n∑i=1

u(yi(ε∗))−

n−1∑i=1

ci(ei)

=n∑i=1

ci(ei)−n−1∑i=1

ci(ei)

= cn(en),

where the third equality follows (12), and the fifth equality follows (11). Thus, agent nalso receives an expected utility exactly equal to his effort cost cn(en). We can concludethat the generalized contest implements e with an expected sum of transfers x′.

Case 2: x′ < x. Let y = (x1, . . . , xn). Consider a generalized contest where in equi-librium, when e = e, each agent i receives xi with probability one, and is thus exactlycompensated for the cost of effort.

Next, we construct the vector of prizes yd that will be used after unilateral deviations.Note that (n−1)u(x/(n−1)) > (n−1)u(x′/(n−1)) =

∑ni=1 ci(ei). In addition, by concavity

of u and u(0) = 0 we have u(x) ≤∑n

i=1 ui(xi) =∑n

i=1 ci(ei). But then by continuity of uthere must exist ε∗ ∈

(0, x

n−1

]such that

(n− 2)u

(x

n− 1− ε∗

)+ u

(x

n− 1+ (n− 2)ε∗

)=

n∑i=1

ci(ei) = (n− 1)u

(x′

n− 1

).

Now define

yd =

(x

n− 1+ (n− 2)ε∗,

x

n− 1− ε∗, . . . , x

n− 1− ε∗, 0

).

Let the zero prize be allocated to a unilateral deviator from e, while the other prizes inyd are allocated randomly among the remaining agents. By construction of y and yd, thisgeneralized contest is credible, and it follows immediately that it implements e, with anexpected sum of transfers x. �

64

Lemma 16 If a pure-strategy effort profile e is implementable, then the lowest expectedsum of transfers at which it can be implemented is max{x′, x}.

Proof. First, observe that any contract implementing e has to satisfy ci(ei) ≤ Eµe [u(ti)] ≤u(Eµe [ti]) and hence xi = u−1(ci(ei)) ≤ Eµe [ti] for all i ∈ I. Summing up over all i ∈ Iimplies that x ≤

∑ni=1 Eµe [ti], so that x is a lower bound on the expected sum of transfers

in any contract implementing e.When x′ < x, the construction given in Case 2 of the proof of Lemma 15 implements e

at exactly x, so the lower bound is reached.When x′ ≥ x, the construction given in Case 1 of the proof of Lemma 15 implements e

at exactly x′. We claim that in this case x′ is in fact the lowest expected sum of transfers atwhich e can be implemented. Suppose by contradiction that there exists a contract whichimplements e at x < x′. By Lemma 7 and the definition of x′, we have

u

(x′

n− 1

)=

1

n− 1

n∑i=1

ci(ei) ≤ u

(x

n− 1

),

which implies x ≥ x′, a contradiction. �

To complete the proof of Proposition 4, we finally establish the existence of a solution.

Lemma 17 An optimal contract exists.

Proof. First, observe that the set of implementable effort profiles E1 ∩ E2 is compact.By Lemma 16, the minimum expected sum of transfers for implementing an effort profilee ∈ E1 ∩ E2 is given by max{x′(e), x(e)}, where

x′(e) = (n− 1)u−1

(1

n− 1

n∑i=1

ci(ei)

)and x(e) =

n∑i=1

u−1(ci(ei))

are continuous functions of e. Thus, the principal’s problem can be reformulated as

maxe∈E1∩E2

[n∑i=1

ei −max{x′(e), x(e)}

],

which amounts to the maximization of a continuous function on a compact set. Hence asolution exists. � �

65